In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point (after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating-point number representation.
Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor has no floating point unit (FPU) or if fixed-point provides improved performance or accuracy for the application at hand. Most low-cost embeddedmicroprocessors and microcontrollers do not have an FPU.
Representation
A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type. For example, the value 1.23 can be represented as 1230 in a fixed-point data type with scaling factor of 1/1000, and the value 1,230,000 can be represented as 1230 with a scaling factor of 1000. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation.
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only iff(c) = c. This means f(f(...f(c)...)) = fn(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.
For example, if f is defined on the real numbers by
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, if f is a function defined on the real numbers as f(x) = x + 1, then it has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheore...
published: 28 Sep 2016
Fixed Point Iteration
Fixed Point Iteration method for finding roots of functions.
Frequently Asked Questions:
Where did 1.618 come from?
If you keep iterating the example will eventually converge on 1.61803398875... which is (1+sqrt(5))/2.
Why not use x = x^2 -1?
Generally you try to reduce the degree of the polynomial you're trying to find the root for.
How did you pick x1?
Your starting point should be an educated guess, a point in the neighborhood of your root.
How can you use the convergence test without the root?
Think of the convergence test as more of "will this function converge to this root?" When you don't know the root, try iterating a few times to see if the function is converging, bouncing around in a loop, or going to infinity. It will become apparent very quickly.
What happens if a function ...
published: 27 Mar 2011
Fixed Point Iteration Method Intuition | Numerical Methods
Let’s talk about the fixed point iteration method, in particular the intuition behind the fixed point method. The fixed point method is an open-root finding method much like some other methods we have learned (Newton’s Method). We find are essentially setting up two functions that when equal will meet at a point in space directly above the root of our function.
This timeline is meant to help you better understand the Fixed Point Method Intuition:
0:00 Introduction.
0:05 What is the fixed-point method?
0:48 Fixed point method intuition
3:08 Outro
Relevant Numerical Methods Playlists:
Numerical Methods Playlist: https://www.youtube.com/watch?v=F2NWYEOHIkc&list=PLDea8VeK4MUTOBXLpvx_WKtVrMkojEh52
Numerical Methods Examples Playlist: https://www.youtube.com/watch?v=UA7bzwCwHMI&list=PLDea8VeK...
published: 07 Feb 2022
Fixed Point Decimal Numbers - Including Fixed Point Arithmetic
Floating point numbers are used a lot in computing from 3D graphics to the latest AI models, they are everywhere! I want to make a video about floating point numbers, but before that I think it is important to cover fixed point numbers. So this video is about fixed point numbers and the next one will be about floating point numbers!
---
Let Me Explain T-shirt: https://teespring.com/gary-explains-let-me-explain-2
Twitter: https://twitter.com/garyexplains
Instagram: https://www.instagram.com/garyexplains/
#garyexplains
published: 17 Apr 2024
Ep 017: Fixed Point Notation Basics
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
published: 21 Aug 2020
🟢09a - Fixed Point Iteration Method (Intro): Example 1
In this lesson, we shall consider the problem of finding the roots or solutions to fixed-point iteration systems.
This video covers the introduction to the topic. In the introduction, we shall solve three different examples. This video covers only Example 1.
00:00 - Example 1
Playlists on various Course
1. Applied Electricity
https://www.youtube.com/playlist?list=PLInywrvFyvq7pFsDEDu2-n0f5UOhpqWBD
2. Linear Algebra / Math 151
https://www.youtube.com/playlist?list=PLInywrvFyvq4IE-nW-ikwkZ2v81n31HQX
3. Basic Mechanics
https://www.youtube.com/playlist?list=PLInywrvFyvq6FUfAigJ3157kg-nZ020fd
4. Calculus with Analysis / Calculus 1 / Math 152
https://www.youtube.com/playlist?list=PLInywrvFyvq6_G3iA7LHbt5exJgGbp4Ok
5. Differential Equations / Math 251
https://www.youtube.com/playlist?list=P...
published: 02 Dec 2023
A Level Computer Science - Fixed Point Binary
How to represent decimal numbers in binary using a fixed point system
published: 18 Mar 2017
Floating Point Numbers | Fixed Point Number vs Floating Point Numbers
In this video, the difference between the Fixed Point and Floating Point Representation is explained. And how very large or very small numbers are stored in a computer using the floating point representation is explained. Also at the end, how to represent any Binary number in Floating Point representation is explained through examples.
Fixed Point Representation:
In Fixed Point Representation, the position of the radix point remains fixed. Because of the fixed radix point/ decimal point, for a given number of bits, the range of Binary numbers gets restricted.
Using 16 bits or 32 bits, in a fixed point representation, it is not possible to represent very large or very small numbers.
Floating Point Representation:
In Floating Point Representation, the position of the radix point / Binary...
published: 25 Nov 2023
Evolution of an attracting basin
The origin z^* = 0 is an attracting fixed point for the Newton function
N(z) = z - (exp(10 z^E) - 1)/(10 E z^(E-1) exp(10 z^E)). The attracting basin od z^* is the set of all points whose orbits converge to z^*. The black region is the exterior of the basin. The different colors depend on the speed of the convergence (in the red and yellow regions this speed is greater than in the regions violet or green).
In this video we will see the evolution of the attracting basin with the exponent E.
published: 09 Sep 2024
Back To Basics! Fixed Point Numbers in C++
In this video I look at a simple Fixed Point Number implementation in C++. I use constexpr to let the IDE run the code without even compiling anything! The final result is a small template class that permits arbitrary precision fixed point calculations.
Patreon: https://www.patreon.com/javidx9
YouTube: https://www.youtube.com/javidx9
https://www.youtube.com/javidx9extra
Discord: https://discord.gg/WhwHUMV
Twitter: https://www.twitter.com/javidx9
Twitch: https://www.twitch.tv/javidx9
GitHub: https://www.github.com/onelonecoder
Homepage: https://www.onelonecoder.com
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My t...
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheorem/
Online graphing calculator used was: https://www.desmos.com/
fixed points more generally:
http://www.math.uri.edu/~kulenm/mth381pr/fixedpoint/fixedpoint.html
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Fixed-point_iteration
Gullible
9s: http://mathforum.org/library/drmath/view/62561.html
Babylonian method:
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
Iterated equation solver you can use to run the Babylonian method: http://www.wolframalpha.com/widgets/view.jsp?id=e4a9b48a942efefa14b2773c31982c29
aleph fixed point:
https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/
http://cantorsattic.info/Aleph
My video on counting past infinity: https://www.youtube.com/watch?v=SrU9YDoXE88
map projections:
http://www.gis.osu.edu/misc/map-projections/
https://www.jasondavies.com/maps/transition/
antipodes: http://geographer-at-large.blogspot.co.uk/2011/10/where-is-your-antipode.html
antipode-finders:
https://www.jasondavies.com/maps/antipodes/
http://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
Pacific antipode image: http://i.imgur.com/dyJQ3EW.jpg
Borsuk-Ulam:
https://www.youtube.com/watch?v=5Px6fajpSio
https://www.youtube.com/watch?v=_sA3c0AhK34
https://www.youtube.com/watch?v=EhAs-vZDQG0
“four” is the cosmic number:
Standupmaths video about how four has four letters. So bummed I only found this AFTER I made this vid, check it out!! https://www.youtube.com/watch?v=LYKn0yUTIU4
http://theweeklyriddle.blogspot.com/2010/09/cosmic-number.html
“Idempotent”: https://en.wikipedia.org/wiki/Idempotence
protocol_7’s comment: https://www.reddit.com/r/math/comments/tw502/can_you_supply_an_elegant_proof/
Other topology pages:
http://www.solitaryroad.com/c775.html
https://en.wikipedia.org/wiki/Homeomorphism
wiki images:
https://commons.wikimedia.org/wiki/File:Robert_Rauschenberg_(1968).jpg
https://en.wikipedia.org/wiki/Claes_Oldenburg#/media/File:Claes_Oldenburg_(1970).jpg
Globe is from Ballerby & Co. Globemakers
http://www.bellerbyandco.com/
https://www.instagram.com/globemakers/
music from
http://www.youtube.com/JakeChudnow
and http://www.audionetwork.com
Any VFX that actually look cool are by https://www.youtube.com/c/ericlanglay -- I did the rest the best I could :)
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheorem/
Online graphing calculator used was: https://www.desmos.com/
fixed points more generally:
http://www.math.uri.edu/~kulenm/mth381pr/fixedpoint/fixedpoint.html
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Fixed-point_iteration
Gullible
9s: http://mathforum.org/library/drmath/view/62561.html
Babylonian method:
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
Iterated equation solver you can use to run the Babylonian method: http://www.wolframalpha.com/widgets/view.jsp?id=e4a9b48a942efefa14b2773c31982c29
aleph fixed point:
https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/
http://cantorsattic.info/Aleph
My video on counting past infinity: https://www.youtube.com/watch?v=SrU9YDoXE88
map projections:
http://www.gis.osu.edu/misc/map-projections/
https://www.jasondavies.com/maps/transition/
antipodes: http://geographer-at-large.blogspot.co.uk/2011/10/where-is-your-antipode.html
antipode-finders:
https://www.jasondavies.com/maps/antipodes/
http://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
Pacific antipode image: http://i.imgur.com/dyJQ3EW.jpg
Borsuk-Ulam:
https://www.youtube.com/watch?v=5Px6fajpSio
https://www.youtube.com/watch?v=_sA3c0AhK34
https://www.youtube.com/watch?v=EhAs-vZDQG0
“four” is the cosmic number:
Standupmaths video about how four has four letters. So bummed I only found this AFTER I made this vid, check it out!! https://www.youtube.com/watch?v=LYKn0yUTIU4
http://theweeklyriddle.blogspot.com/2010/09/cosmic-number.html
“Idempotent”: https://en.wikipedia.org/wiki/Idempotence
protocol_7’s comment: https://www.reddit.com/r/math/comments/tw502/can_you_supply_an_elegant_proof/
Other topology pages:
http://www.solitaryroad.com/c775.html
https://en.wikipedia.org/wiki/Homeomorphism
wiki images:
https://commons.wikimedia.org/wiki/File:Robert_Rauschenberg_(1968).jpg
https://en.wikipedia.org/wiki/Claes_Oldenburg#/media/File:Claes_Oldenburg_(1970).jpg
Globe is from Ballerby & Co. Globemakers
http://www.bellerbyandco.com/
https://www.instagram.com/globemakers/
music from
http://www.youtube.com/JakeChudnow
and http://www.audionetwork.com
Any VFX that actually look cool are by https://www.youtube.com/c/ericlanglay -- I did the rest the best I could :)
Fixed Point Iteration method for finding roots of functions.
Frequently Asked Questions:
Where did 1.618 come from?
If you keep iterating the example will event...
Fixed Point Iteration method for finding roots of functions.
Frequently Asked Questions:
Where did 1.618 come from?
If you keep iterating the example will eventually converge on 1.61803398875... which is (1+sqrt(5))/2.
Why not use x = x^2 -1?
Generally you try to reduce the degree of the polynomial you're trying to find the root for.
How did you pick x1?
Your starting point should be an educated guess, a point in the neighborhood of your root.
How can you use the convergence test without the root?
Think of the convergence test as more of "will this function converge to this root?" When you don't know the root, try iterating a few times to see if the function is converging, bouncing around in a loop, or going to infinity. It will become apparent very quickly.
What happens if a function fails the convergence test?
Failing the test means that the function is not guaranteed to converge. It might still converge but it makes no promises. Take the function which I showed fail in the example. If you iterate starting from the root that we found, the function might converge to the same value depending on your calculator's accuracy.
Doesn't this function have two roots? Is there a way to find the second one?
Indeed this function has two roots (1+sqrt(5))/2 and (1-sqrt(5))/2 which are the numbers φ (phi) and ψ (psi). I showed how the first example converged to phi and that the other did not for simplicity. You can use the second equation to converge on psi if you start close enough, like -1 for example.
Is there any way to use x = +/- sqrt(x + 1)?
In this case you can use x = sqrt(x+1) which will converge to 1.618 as long as the value inside the square root is positive. If you try to take the square root of a negative number you will have to use imaginary and complex numbers.
Is there a way to speed up Fixed Point Iteration?
Yes, check out my video on Steffensen's Method with Aitken's Δ² https://youtu.be/BTYTj0r5PZE and my video on Wegstein's Method https://youtu.be/T_6mR6rJXQQ
How can I force Fixed Point Iteration to converge?
There is a very simple change you can make to induce convergence called Wegstein's Method https://youtu.be/T_6mR6rJXQQ
Can you make a video that answers these questions?
Absolutely check out Fixed Point Iteration Q&A https://youtu.be/FyCviw2ZA2o
Chapters
0:00 Intro
0:06 Fixed Point Iteration
0:39 Fixed Point Iteration Example
2:12 Convergence Test
2:41 Convergence Test Example
3:18 Order
4:03 Thanks For Watching
Further Viewing:
Fixed Point Iteration Q&A https://youtu.be/FyCviw2ZA2o
Steffensen's Method with Aitken's Δ² https://youtu.be/BTYTj0r5PZE
Wegstein's Method https://youtu.be/T_6mR6rJXQQ
Fixed Point Iteration Systems of Equations https://youtu.be/xa2vUsYJD-c
Generalized Aitken-Steffensen Method https://youtu.be/-x9_fSNrX3w
#FixedPointIteration #NumericalAnalysis
Fixed Point Iteration method for finding roots of functions.
Frequently Asked Questions:
Where did 1.618 come from?
If you keep iterating the example will eventually converge on 1.61803398875... which is (1+sqrt(5))/2.
Why not use x = x^2 -1?
Generally you try to reduce the degree of the polynomial you're trying to find the root for.
How did you pick x1?
Your starting point should be an educated guess, a point in the neighborhood of your root.
How can you use the convergence test without the root?
Think of the convergence test as more of "will this function converge to this root?" When you don't know the root, try iterating a few times to see if the function is converging, bouncing around in a loop, or going to infinity. It will become apparent very quickly.
What happens if a function fails the convergence test?
Failing the test means that the function is not guaranteed to converge. It might still converge but it makes no promises. Take the function which I showed fail in the example. If you iterate starting from the root that we found, the function might converge to the same value depending on your calculator's accuracy.
Doesn't this function have two roots? Is there a way to find the second one?
Indeed this function has two roots (1+sqrt(5))/2 and (1-sqrt(5))/2 which are the numbers φ (phi) and ψ (psi). I showed how the first example converged to phi and that the other did not for simplicity. You can use the second equation to converge on psi if you start close enough, like -1 for example.
Is there any way to use x = +/- sqrt(x + 1)?
In this case you can use x = sqrt(x+1) which will converge to 1.618 as long as the value inside the square root is positive. If you try to take the square root of a negative number you will have to use imaginary and complex numbers.
Is there a way to speed up Fixed Point Iteration?
Yes, check out my video on Steffensen's Method with Aitken's Δ² https://youtu.be/BTYTj0r5PZE and my video on Wegstein's Method https://youtu.be/T_6mR6rJXQQ
How can I force Fixed Point Iteration to converge?
There is a very simple change you can make to induce convergence called Wegstein's Method https://youtu.be/T_6mR6rJXQQ
Can you make a video that answers these questions?
Absolutely check out Fixed Point Iteration Q&A https://youtu.be/FyCviw2ZA2o
Chapters
0:00 Intro
0:06 Fixed Point Iteration
0:39 Fixed Point Iteration Example
2:12 Convergence Test
2:41 Convergence Test Example
3:18 Order
4:03 Thanks For Watching
Further Viewing:
Fixed Point Iteration Q&A https://youtu.be/FyCviw2ZA2o
Steffensen's Method with Aitken's Δ² https://youtu.be/BTYTj0r5PZE
Wegstein's Method https://youtu.be/T_6mR6rJXQQ
Fixed Point Iteration Systems of Equations https://youtu.be/xa2vUsYJD-c
Generalized Aitken-Steffensen Method https://youtu.be/-x9_fSNrX3w
#FixedPointIteration #NumericalAnalysis
Let’s talk about the fixed point iteration method, in particular the intuition behind the fixed point method. The fixed point method is an open-root finding met...
Let’s talk about the fixed point iteration method, in particular the intuition behind the fixed point method. The fixed point method is an open-root finding method much like some other methods we have learned (Newton’s Method). We find are essentially setting up two functions that when equal will meet at a point in space directly above the root of our function.
This timeline is meant to help you better understand the Fixed Point Method Intuition:
0:00 Introduction.
0:05 What is the fixed-point method?
0:48 Fixed point method intuition
3:08 Outro
Relevant Numerical Methods Playlists:
Numerical Methods Playlist: https://www.youtube.com/watch?v=F2NWYEOHIkc&list=PLDea8VeK4MUTOBXLpvx_WKtVrMkojEh52
Numerical Methods Examples Playlist: https://www.youtube.com/watch?v=UA7bzwCwHMI&list=PLDea8VeK4MUTxZGOTG-VNnW4T3TYwlEc9
Follow & Support StudySession:
Channel Memberships: https://www.youtube.com/channel/UCaFHlSOg83nCUIHlFMlUhPw/join
Email Us: [email protected]
Twitter: https://twitter.com/StudySessionYT
Instagram: https://instagram.com/StudySessionyt/
This video is part of our Numerical Methods course. Numerical methods is about solving math problems through approximating the solution of problems that would be difficult or impossible to solve analytically. In this playlist we will cover topics such as solving systems of linear equations, solving systems of non-linear equations, numerical integration, numerical derivatives, etc..
#studysession #numericalmethods #fixedpoint
Let’s talk about the fixed point iteration method, in particular the intuition behind the fixed point method. The fixed point method is an open-root finding method much like some other methods we have learned (Newton’s Method). We find are essentially setting up two functions that when equal will meet at a point in space directly above the root of our function.
This timeline is meant to help you better understand the Fixed Point Method Intuition:
0:00 Introduction.
0:05 What is the fixed-point method?
0:48 Fixed point method intuition
3:08 Outro
Relevant Numerical Methods Playlists:
Numerical Methods Playlist: https://www.youtube.com/watch?v=F2NWYEOHIkc&list=PLDea8VeK4MUTOBXLpvx_WKtVrMkojEh52
Numerical Methods Examples Playlist: https://www.youtube.com/watch?v=UA7bzwCwHMI&list=PLDea8VeK4MUTxZGOTG-VNnW4T3TYwlEc9
Follow & Support StudySession:
Channel Memberships: https://www.youtube.com/channel/UCaFHlSOg83nCUIHlFMlUhPw/join
Email Us: [email protected]
Twitter: https://twitter.com/StudySessionYT
Instagram: https://instagram.com/StudySessionyt/
This video is part of our Numerical Methods course. Numerical methods is about solving math problems through approximating the solution of problems that would be difficult or impossible to solve analytically. In this playlist we will cover topics such as solving systems of linear equations, solving systems of non-linear equations, numerical integration, numerical derivatives, etc..
#studysession #numericalmethods #fixedpoint
Floating point numbers are used a lot in computing from 3D graphics to the latest AI models, they are everywhere! I want to make a video about floating point nu...
Floating point numbers are used a lot in computing from 3D graphics to the latest AI models, they are everywhere! I want to make a video about floating point numbers, but before that I think it is important to cover fixed point numbers. So this video is about fixed point numbers and the next one will be about floating point numbers!
---
Let Me Explain T-shirt: https://teespring.com/gary-explains-let-me-explain-2
Twitter: https://twitter.com/garyexplains
Instagram: https://www.instagram.com/garyexplains/
#garyexplains
Floating point numbers are used a lot in computing from 3D graphics to the latest AI models, they are everywhere! I want to make a video about floating point numbers, but before that I think it is important to cover fixed point numbers. So this video is about fixed point numbers and the next one will be about floating point numbers!
---
Let Me Explain T-shirt: https://teespring.com/gary-explains-let-me-explain-2
Twitter: https://twitter.com/garyexplains
Instagram: https://www.instagram.com/garyexplains/
#garyexplains
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned bi...
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
In this lesson, we shall consider the problem of finding the roots or solutions to fixed-point iteration systems.
This video covers the introduction to the topi...
In this lesson, we shall consider the problem of finding the roots or solutions to fixed-point iteration systems.
This video covers the introduction to the topic. In the introduction, we shall solve three different examples. This video covers only Example 1.
00:00 - Example 1
Playlists on various Course
1. Applied Electricity
https://www.youtube.com/playlist?list=PLInywrvFyvq7pFsDEDu2-n0f5UOhpqWBD
2. Linear Algebra / Math 151
https://www.youtube.com/playlist?list=PLInywrvFyvq4IE-nW-ikwkZ2v81n31HQX
3. Basic Mechanics
https://www.youtube.com/playlist?list=PLInywrvFyvq6FUfAigJ3157kg-nZ020fd
4. Calculus with Analysis / Calculus 1 / Math 152
https://www.youtube.com/playlist?list=PLInywrvFyvq6_G3iA7LHbt5exJgGbp4Ok
5. Differential Equations / Math 251
https://www.youtube.com/playlist?list=PLInywrvFyvq408vWA5OYXShA6rlT51TdS
6. Electric Circuit Theory / Circuit Design
https://www.youtube.com/playlist?list=PLInywrvFyvq4sNicTbLBUpgkxrkcs2OGN
7. Calculus with Several Variables
https://www.youtube.com/playlist?list=PLInywrvFyvq5IXmWYBTTMvM_EdNcfUVt3
8. Numerical Analysis
https://www.youtube.com/playlist?list=PLInywrvFyvq7oAlPscVnXsd8CRTsh0b77
Make sure to watch till the end.
Like, share, and subscribe.
Thank you.
In this lesson, we shall consider the problem of finding the roots or solutions to fixed-point iteration systems.
This video covers the introduction to the topic. In the introduction, we shall solve three different examples. This video covers only Example 1.
00:00 - Example 1
Playlists on various Course
1. Applied Electricity
https://www.youtube.com/playlist?list=PLInywrvFyvq7pFsDEDu2-n0f5UOhpqWBD
2. Linear Algebra / Math 151
https://www.youtube.com/playlist?list=PLInywrvFyvq4IE-nW-ikwkZ2v81n31HQX
3. Basic Mechanics
https://www.youtube.com/playlist?list=PLInywrvFyvq6FUfAigJ3157kg-nZ020fd
4. Calculus with Analysis / Calculus 1 / Math 152
https://www.youtube.com/playlist?list=PLInywrvFyvq6_G3iA7LHbt5exJgGbp4Ok
5. Differential Equations / Math 251
https://www.youtube.com/playlist?list=PLInywrvFyvq408vWA5OYXShA6rlT51TdS
6. Electric Circuit Theory / Circuit Design
https://www.youtube.com/playlist?list=PLInywrvFyvq4sNicTbLBUpgkxrkcs2OGN
7. Calculus with Several Variables
https://www.youtube.com/playlist?list=PLInywrvFyvq5IXmWYBTTMvM_EdNcfUVt3
8. Numerical Analysis
https://www.youtube.com/playlist?list=PLInywrvFyvq7oAlPscVnXsd8CRTsh0b77
Make sure to watch till the end.
Like, share, and subscribe.
Thank you.
In this video, the difference between the Fixed Point and Floating Point Representation is explained. And how very large or very small numbers are stored in a c...
In this video, the difference between the Fixed Point and Floating Point Representation is explained. And how very large or very small numbers are stored in a computer using the floating point representation is explained. Also at the end, how to represent any Binary number in Floating Point representation is explained through examples.
Fixed Point Representation:
In Fixed Point Representation, the position of the radix point remains fixed. Because of the fixed radix point/ decimal point, for a given number of bits, the range of Binary numbers gets restricted.
Using 16 bits or 32 bits, in a fixed point representation, it is not possible to represent very large or very small numbers.
Floating Point Representation:
In Floating Point Representation, the position of the radix point / Binary Point is dynamic. Therefore, it is possible to represent very large or very small numbers.
The floating Point Number representation is similar to the scientific notation used for the decimal numbers. It is a normalized representation.
In Floating Point Representation, there is only a significant digit before the Binary point.
The floating Point Representation consists of 3 parts.
1) Sign
2) Significand
3) Exponent
In this video, how to represent any Binary Number in a floating point representation is explained using the examples.
Check out the other useful videos related to Digital Electronics (Playlist):
https://youtube.com/playlist?list=PLwjK_iyK4LLBC_so3odA64E2MLgIRKafl&si=wpYsuM0VFsvMwftG
The following topics are covered in the video:
0:00 Introduction
0:44 Fixed Point Numbers and their limitation
6:41 Floating Point Number Representation
This video will be helpful to all the students of science and engineering in understanding how very large or very small numbers are stored in a computer using the floating point number representation.
#allaboutelectronics
#digitalelectronics
#floatingpointnumbers
#numbersystem
Support the channel through membership program:
https://www.youtube.com/channel/UCBkOVp1Cqz4MR0LYR8vKpZg/join
--------------------------------------------------------------------------------------------------
Follow my second channel:
https://www.youtube.com/channel/UCGA2TO8ylVqFHpucwn_6Jlw
Follow me on Facebook:
https://www.facebook.com/ALLABOUTELECRONICS/
Follow me on Instagram:
https://www.instagram.com/all_about.electronics/
--------------------------------------------------------------------------------------------------
Music Credit: http://www.bensound.com
In this video, the difference between the Fixed Point and Floating Point Representation is explained. And how very large or very small numbers are stored in a computer using the floating point representation is explained. Also at the end, how to represent any Binary number in Floating Point representation is explained through examples.
Fixed Point Representation:
In Fixed Point Representation, the position of the radix point remains fixed. Because of the fixed radix point/ decimal point, for a given number of bits, the range of Binary numbers gets restricted.
Using 16 bits or 32 bits, in a fixed point representation, it is not possible to represent very large or very small numbers.
Floating Point Representation:
In Floating Point Representation, the position of the radix point / Binary Point is dynamic. Therefore, it is possible to represent very large or very small numbers.
The floating Point Number representation is similar to the scientific notation used for the decimal numbers. It is a normalized representation.
In Floating Point Representation, there is only a significant digit before the Binary point.
The floating Point Representation consists of 3 parts.
1) Sign
2) Significand
3) Exponent
In this video, how to represent any Binary Number in a floating point representation is explained using the examples.
Check out the other useful videos related to Digital Electronics (Playlist):
https://youtube.com/playlist?list=PLwjK_iyK4LLBC_so3odA64E2MLgIRKafl&si=wpYsuM0VFsvMwftG
The following topics are covered in the video:
0:00 Introduction
0:44 Fixed Point Numbers and their limitation
6:41 Floating Point Number Representation
This video will be helpful to all the students of science and engineering in understanding how very large or very small numbers are stored in a computer using the floating point number representation.
#allaboutelectronics
#digitalelectronics
#floatingpointnumbers
#numbersystem
Support the channel through membership program:
https://www.youtube.com/channel/UCBkOVp1Cqz4MR0LYR8vKpZg/join
--------------------------------------------------------------------------------------------------
Follow my second channel:
https://www.youtube.com/channel/UCGA2TO8ylVqFHpucwn_6Jlw
Follow me on Facebook:
https://www.facebook.com/ALLABOUTELECRONICS/
Follow me on Instagram:
https://www.instagram.com/all_about.electronics/
--------------------------------------------------------------------------------------------------
Music Credit: http://www.bensound.com
The origin z^* = 0 is an attracting fixed point for the Newton function
N(z) = z - (exp(10 z^E) - 1)/(10 E z^(E-1) exp(10 z^E)). The attracting basin od z^* is ...
The origin z^* = 0 is an attracting fixed point for the Newton function
N(z) = z - (exp(10 z^E) - 1)/(10 E z^(E-1) exp(10 z^E)). The attracting basin od z^* is the set of all points whose orbits converge to z^*. The black region is the exterior of the basin. The different colors depend on the speed of the convergence (in the red and yellow regions this speed is greater than in the regions violet or green).
In this video we will see the evolution of the attracting basin with the exponent E.
The origin z^* = 0 is an attracting fixed point for the Newton function
N(z) = z - (exp(10 z^E) - 1)/(10 E z^(E-1) exp(10 z^E)). The attracting basin od z^* is the set of all points whose orbits converge to z^*. The black region is the exterior of the basin. The different colors depend on the speed of the convergence (in the red and yellow regions this speed is greater than in the regions violet or green).
In this video we will see the evolution of the attracting basin with the exponent E.
In this video I look at a simple Fixed Point Number implementation in C++. I use constexpr to let the IDE run the code without even compiling anything! The fina...
In this video I look at a simple Fixed Point Number implementation in C++. I use constexpr to let the IDE run the code without even compiling anything! The final result is a small template class that permits arbitrary precision fixed point calculations.
Patreon: https://www.patreon.com/javidx9
YouTube: https://www.youtube.com/javidx9
https://www.youtube.com/javidx9extra
Discord: https://discord.gg/WhwHUMV
Twitter: https://www.twitter.com/javidx9
Twitch: https://www.twitch.tv/javidx9
GitHub: https://www.github.com/onelonecoder
Homepage: https://www.onelonecoder.com
In this video I look at a simple Fixed Point Number implementation in C++. I use constexpr to let the IDE run the code without even compiling anything! The final result is a small template class that permits arbitrary precision fixed point calculations.
Patreon: https://www.patreon.com/javidx9
YouTube: https://www.youtube.com/javidx9
https://www.youtube.com/javidx9extra
Discord: https://discord.gg/WhwHUMV
Twitter: https://www.twitter.com/javidx9
Twitch: https://www.twitch.tv/javidx9
GitHub: https://www.github.com/onelonecoder
Homepage: https://www.onelonecoder.com
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
published: 21 Aug 2020
Fixed Points
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheore...
published: 28 Sep 2016
Floating Point Numbers (Part1: Fp vs Fixed) - Computerphile
How much does a floating point processor improve floating point operations? Dr Bagley installed one to find out - and explains how computers store the bits.
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
published: 01 Mar 2019
Find the Fixed Points of a Function
✔ https://StudyForce.com
✔ https://Biology-Forums.com
✔ Ask questions here: https://Biology-Forums.com/index.php?board=33.0
Follow us:
▶ Facebook: https://facebook.com/StudyForcePS/
▶ Instagram: https://instagram.com/biologyforums/
▶ Twitter: https://twitter.com/studyforceps
Note: To find the fix points of a function means to find when the input values equal the output values. Since the equation y=x is purely a one-to-one function, to find any fixed points for any function, we set that function equal to x and solve.
Q1. Find the fixed points of f(x)=∛x
Q2. Find the fixed points of h(x)=x^2−3x
Q3. Find the fixed points of the second iterate of g(x)=x^2−1
published: 03 Nov 2022
Fixed Point Maths Explained - Retro Programming
A video explaining how fixed point maths works and why it is useful on CPUs that have no floating point units.
A written version of this is available on my website, linked below
Fixed point maths is a way to deal with decimal numbers, on systems where they don't have the ability to use decimals. The basic concept is to scale all numbers up by a certain amount to remove any decimal parts of the number. Then you can work with whole numbers only, scaling back down when needed.
What scale you use depends on the range of numbers needed for a particular situation. It can also depend on what the hardware is best at managing. A Z80 CPU for example can work with 16 bit values reasonably easily. And if you’re not trying to do complex accurate scientific calculations, the resolution of your machin...
published: 14 Nov 2021
Fixed Point Arithmetic 1: Intro to Fixed Point
In this video we'll look at fixed point arithmetic. This is a technique for performing operations on numbers with fractional parts using integers, instead of the more common floating point.
Fixed point has some pretty amazing benefits, depending on how and when it is used. We can use bytes, short ints, ints, or long integers, we can select the precision and range of our variables.
Fixed point is very flexible and (sometimes) very fast. It's extremely good at certain types of tasks, for instance, working with image processing matrices. It can also do some things which floating point cannot, such as exactly represent 1/3 or 1/5.
Fixed point is not a hack, it's a fantastic tool to add to your belt if you've not yet come across it. If you have used fixed point in the past, I hope this tute ...
published: 25 Mar 2014
Introduction to Fixed Point Math for Embedded Systems - Part 1 of 3
Part 1 of 3
published: 28 Apr 2014
Fixed-point iteration method - convergence and the Fixed-point theorem
In this video, we look at the convergence of the method and its relation to the Fixed-point theorem. Please note there is a mistake at the end of the video 1/x-1 is not less than 1 and the iteration diverges. I will update the video.
published: 27 Sep 2017
Economics A Level Livestream | Costs and Revenues
Join Geoff, Jon and the team for a 30 minute fast-paced revision blast on Costs and Revenues.
published: 08 Nov 2023
Fixed-Point Made Easy for FPGA Programming
One of the biggest challenges in FPGA programming is the process of quantizing mathematical operations to fixed-point for more efficient implementation.
This session teaches the fundamentals of the fixed-point number system and fixed-point arithmetic, along with considerations for targeting popular FPGA devices. These concepts are then reinforced through practical demonstrations, capped by walking through the process of quantizing a signal processing design.
Topics include:
Fixed-point theory
- Fixed-point number system
- Mathematical range
- Quantization error in the time and frequency domains
Common functions
- Arithmetic: square root, reciprocal, log2
- Trigonometry: cosine, sine, atan2
Signal processing: FIR, FFT
- FPGA considerations
- Targeting Xilinx and Intel devices
- Maintaini...
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned bi...
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My t...
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheorem/
Online graphing calculator used was: https://www.desmos.com/
fixed points more generally:
http://www.math.uri.edu/~kulenm/mth381pr/fixedpoint/fixedpoint.html
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Fixed-point_iteration
Gullible
9s: http://mathforum.org/library/drmath/view/62561.html
Babylonian method:
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
Iterated equation solver you can use to run the Babylonian method: http://www.wolframalpha.com/widgets/view.jsp?id=e4a9b48a942efefa14b2773c31982c29
aleph fixed point:
https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/
http://cantorsattic.info/Aleph
My video on counting past infinity: https://www.youtube.com/watch?v=SrU9YDoXE88
map projections:
http://www.gis.osu.edu/misc/map-projections/
https://www.jasondavies.com/maps/transition/
antipodes: http://geographer-at-large.blogspot.co.uk/2011/10/where-is-your-antipode.html
antipode-finders:
https://www.jasondavies.com/maps/antipodes/
http://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
Pacific antipode image: http://i.imgur.com/dyJQ3EW.jpg
Borsuk-Ulam:
https://www.youtube.com/watch?v=5Px6fajpSio
https://www.youtube.com/watch?v=_sA3c0AhK34
https://www.youtube.com/watch?v=EhAs-vZDQG0
“four” is the cosmic number:
Standupmaths video about how four has four letters. So bummed I only found this AFTER I made this vid, check it out!! https://www.youtube.com/watch?v=LYKn0yUTIU4
http://theweeklyriddle.blogspot.com/2010/09/cosmic-number.html
“Idempotent”: https://en.wikipedia.org/wiki/Idempotence
protocol_7’s comment: https://www.reddit.com/r/math/comments/tw502/can_you_supply_an_elegant_proof/
Other topology pages:
http://www.solitaryroad.com/c775.html
https://en.wikipedia.org/wiki/Homeomorphism
wiki images:
https://commons.wikimedia.org/wiki/File:Robert_Rauschenberg_(1968).jpg
https://en.wikipedia.org/wiki/Claes_Oldenburg#/media/File:Claes_Oldenburg_(1970).jpg
Globe is from Ballerby & Co. Globemakers
http://www.bellerbyandco.com/
https://www.instagram.com/globemakers/
music from
http://www.youtube.com/JakeChudnow
and http://www.audionetwork.com
Any VFX that actually look cool are by https://www.youtube.com/c/ericlanglay -- I did the rest the best I could :)
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheorem/
Online graphing calculator used was: https://www.desmos.com/
fixed points more generally:
http://www.math.uri.edu/~kulenm/mth381pr/fixedpoint/fixedpoint.html
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Fixed-point_iteration
Gullible
9s: http://mathforum.org/library/drmath/view/62561.html
Babylonian method:
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
Iterated equation solver you can use to run the Babylonian method: http://www.wolframalpha.com/widgets/view.jsp?id=e4a9b48a942efefa14b2773c31982c29
aleph fixed point:
https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/
http://cantorsattic.info/Aleph
My video on counting past infinity: https://www.youtube.com/watch?v=SrU9YDoXE88
map projections:
http://www.gis.osu.edu/misc/map-projections/
https://www.jasondavies.com/maps/transition/
antipodes: http://geographer-at-large.blogspot.co.uk/2011/10/where-is-your-antipode.html
antipode-finders:
https://www.jasondavies.com/maps/antipodes/
http://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
Pacific antipode image: http://i.imgur.com/dyJQ3EW.jpg
Borsuk-Ulam:
https://www.youtube.com/watch?v=5Px6fajpSio
https://www.youtube.com/watch?v=_sA3c0AhK34
https://www.youtube.com/watch?v=EhAs-vZDQG0
“four” is the cosmic number:
Standupmaths video about how four has four letters. So bummed I only found this AFTER I made this vid, check it out!! https://www.youtube.com/watch?v=LYKn0yUTIU4
http://theweeklyriddle.blogspot.com/2010/09/cosmic-number.html
“Idempotent”: https://en.wikipedia.org/wiki/Idempotence
protocol_7’s comment: https://www.reddit.com/r/math/comments/tw502/can_you_supply_an_elegant_proof/
Other topology pages:
http://www.solitaryroad.com/c775.html
https://en.wikipedia.org/wiki/Homeomorphism
wiki images:
https://commons.wikimedia.org/wiki/File:Robert_Rauschenberg_(1968).jpg
https://en.wikipedia.org/wiki/Claes_Oldenburg#/media/File:Claes_Oldenburg_(1970).jpg
Globe is from Ballerby & Co. Globemakers
http://www.bellerbyandco.com/
https://www.instagram.com/globemakers/
music from
http://www.youtube.com/JakeChudnow
and http://www.audionetwork.com
Any VFX that actually look cool are by https://www.youtube.com/c/ericlanglay -- I did the rest the best I could :)
How much does a floating point processor improve floating point operations? Dr Bagley installed one to find out - and explains how computers store the bits.
h...
How much does a floating point processor improve floating point operations? Dr Bagley installed one to find out - and explains how computers store the bits.
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
How much does a floating point processor improve floating point operations? Dr Bagley installed one to find out - and explains how computers store the bits.
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
✔ https://StudyForce.com
✔ https://Biology-Forums.com
✔ Ask questions here: https://Biology-Forums.com/index.php?board=33.0
Follow us:
▶ Facebook: https://facebook.com/StudyForcePS/
▶ Instagram: https://instagram.com/biologyforums/
▶ Twitter: https://twitter.com/studyforceps
Note: To find the fix points of a function means to find when the input values equal the output values. Since the equation y=x is purely a one-to-one function, to find any fixed points for any function, we set that function equal to x and solve.
Q1. Find the fixed points of f(x)=∛x
Q2. Find the fixed points of h(x)=x^2−3x
Q3. Find the fixed points of the second iterate of g(x)=x^2−1
✔ https://StudyForce.com
✔ https://Biology-Forums.com
✔ Ask questions here: https://Biology-Forums.com/index.php?board=33.0
Follow us:
▶ Facebook: https://facebook.com/StudyForcePS/
▶ Instagram: https://instagram.com/biologyforums/
▶ Twitter: https://twitter.com/studyforceps
Note: To find the fix points of a function means to find when the input values equal the output values. Since the equation y=x is purely a one-to-one function, to find any fixed points for any function, we set that function equal to x and solve.
Q1. Find the fixed points of f(x)=∛x
Q2. Find the fixed points of h(x)=x^2−3x
Q3. Find the fixed points of the second iterate of g(x)=x^2−1
A video explaining how fixed point maths works and why it is useful on CPUs that have no floating point units.
A written version of this is available on my web...
A video explaining how fixed point maths works and why it is useful on CPUs that have no floating point units.
A written version of this is available on my website, linked below
Fixed point maths is a way to deal with decimal numbers, on systems where they don't have the ability to use decimals. The basic concept is to scale all numbers up by a certain amount to remove any decimal parts of the number. Then you can work with whole numbers only, scaling back down when needed.
What scale you use depends on the range of numbers needed for a particular situation. It can also depend on what the hardware is best at managing. A Z80 CPU for example can work with 16 bit values reasonably easily. And if you’re not trying to do complex accurate scientific calculations, the resolution of your machine’s screen is likely to be a factor as well.
But if you want a quick one to use, this explanation uses what is called 8.8 fixed numbers. That is a 16 bit value where the upper 8 bits are the whole number and the lower 8 bits the fractional part. With an imaginary binary point in the middle.
Link to website: https://docs.ncot.uk/computing-maths/fixed-point-arithmetic/index.html
Chapters
00:17 Maths is hard
00:56 Floating point maths
01:45 I have more videos!
03:53 What we're trying to do
04:57 Example with money
06:39 Introducing fixed point maths
08:28 negative numbers
10:39 Explaining fixed point
15:03 Summary
A video explaining how fixed point maths works and why it is useful on CPUs that have no floating point units.
A written version of this is available on my website, linked below
Fixed point maths is a way to deal with decimal numbers, on systems where they don't have the ability to use decimals. The basic concept is to scale all numbers up by a certain amount to remove any decimal parts of the number. Then you can work with whole numbers only, scaling back down when needed.
What scale you use depends on the range of numbers needed for a particular situation. It can also depend on what the hardware is best at managing. A Z80 CPU for example can work with 16 bit values reasonably easily. And if you’re not trying to do complex accurate scientific calculations, the resolution of your machine’s screen is likely to be a factor as well.
But if you want a quick one to use, this explanation uses what is called 8.8 fixed numbers. That is a 16 bit value where the upper 8 bits are the whole number and the lower 8 bits the fractional part. With an imaginary binary point in the middle.
Link to website: https://docs.ncot.uk/computing-maths/fixed-point-arithmetic/index.html
Chapters
00:17 Maths is hard
00:56 Floating point maths
01:45 I have more videos!
03:53 What we're trying to do
04:57 Example with money
06:39 Introducing fixed point maths
08:28 negative numbers
10:39 Explaining fixed point
15:03 Summary
In this video we'll look at fixed point arithmetic. This is a technique for performing operations on numbers with fractional parts using integers, instead of th...
In this video we'll look at fixed point arithmetic. This is a technique for performing operations on numbers with fractional parts using integers, instead of the more common floating point.
Fixed point has some pretty amazing benefits, depending on how and when it is used. We can use bytes, short ints, ints, or long integers, we can select the precision and range of our variables.
Fixed point is very flexible and (sometimes) very fast. It's extremely good at certain types of tasks, for instance, working with image processing matrices. It can also do some things which floating point cannot, such as exactly represent 1/3 or 1/5.
Fixed point is not a hack, it's a fantastic tool to add to your belt if you've not yet come across it. If you have used fixed point in the past, I hope this tute is a good refresher!
Also, I forgot to mention something in the tute:
The technique I used to get the fractional part of the fixed point number literally gives us the bits that comprise the fractional part of the number. If your number is negative, then they will not be the correct bits if you print the fixed point to screen.
Facebook:
https://www.facebook.com/pages/WhatsaCreel/167732956665435
In this video we'll look at fixed point arithmetic. This is a technique for performing operations on numbers with fractional parts using integers, instead of the more common floating point.
Fixed point has some pretty amazing benefits, depending on how and when it is used. We can use bytes, short ints, ints, or long integers, we can select the precision and range of our variables.
Fixed point is very flexible and (sometimes) very fast. It's extremely good at certain types of tasks, for instance, working with image processing matrices. It can also do some things which floating point cannot, such as exactly represent 1/3 or 1/5.
Fixed point is not a hack, it's a fantastic tool to add to your belt if you've not yet come across it. If you have used fixed point in the past, I hope this tute is a good refresher!
Also, I forgot to mention something in the tute:
The technique I used to get the fractional part of the fixed point number literally gives us the bits that comprise the fractional part of the number. If your number is negative, then they will not be the correct bits if you print the fixed point to screen.
Facebook:
https://www.facebook.com/pages/WhatsaCreel/167732956665435
In this video, we look at the convergence of the method and its relation to the Fixed-point theorem. Please note there is a mistake at the end of the video 1/x...
In this video, we look at the convergence of the method and its relation to the Fixed-point theorem. Please note there is a mistake at the end of the video 1/x-1 is not less than 1 and the iteration diverges. I will update the video.
In this video, we look at the convergence of the method and its relation to the Fixed-point theorem. Please note there is a mistake at the end of the video 1/x-1 is not less than 1 and the iteration diverges. I will update the video.
One of the biggest challenges in FPGA programming is the process of quantizing mathematical operations to fixed-point for more efficient implementation.
This s...
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheorem/
Online graphing calculator used was: https://www.desmos.com/
fixed points more generally:
http://www.math.uri.edu/~kulenm/mth381pr/fixedpoint/fixedpoint.html
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Fixed-point_iteration
Gullible
9s: http://mathforum.org/library/drmath/view/62561.html
Babylonian method:
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
Iterated equation solver you can use to run the Babylonian method: http://www.wolframalpha.com/widgets/view.jsp?id=e4a9b48a942efefa14b2773c31982c29
aleph fixed point:
https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/
http://cantorsattic.info/Aleph
My video on counting past infinity: https://www.youtube.com/watch?v=SrU9YDoXE88
map projections:
http://www.gis.osu.edu/misc/map-projections/
https://www.jasondavies.com/maps/transition/
antipodes: http://geographer-at-large.blogspot.co.uk/2011/10/where-is-your-antipode.html
antipode-finders:
https://www.jasondavies.com/maps/antipodes/
http://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
Pacific antipode image: http://i.imgur.com/dyJQ3EW.jpg
Borsuk-Ulam:
https://www.youtube.com/watch?v=5Px6fajpSio
https://www.youtube.com/watch?v=_sA3c0AhK34
https://www.youtube.com/watch?v=EhAs-vZDQG0
“four” is the cosmic number:
Standupmaths video about how four has four letters. So bummed I only found this AFTER I made this vid, check it out!! https://www.youtube.com/watch?v=LYKn0yUTIU4
http://theweeklyriddle.blogspot.com/2010/09/cosmic-number.html
“Idempotent”: https://en.wikipedia.org/wiki/Idempotence
protocol_7’s comment: https://www.reddit.com/r/math/comments/tw502/can_you_supply_an_elegant_proof/
Other topology pages:
http://www.solitaryroad.com/c775.html
https://en.wikipedia.org/wiki/Homeomorphism
wiki images:
https://commons.wikimedia.org/wiki/File:Robert_Rauschenberg_(1968).jpg
https://en.wikipedia.org/wiki/Claes_Oldenburg#/media/File:Claes_Oldenburg_(1970).jpg
Globe is from Ballerby & Co. Globemakers
http://www.bellerbyandco.com/
https://www.instagram.com/globemakers/
music from
http://www.youtube.com/JakeChudnow
and http://www.audionetwork.com
Any VFX that actually look cool are by https://www.youtube.com/c/ericlanglay -- I did the rest the best I could :)
Fixed Point Iteration method for finding roots of functions.
Frequently Asked Questions:
Where did 1.618 come from?
If you keep iterating the example will eventually converge on 1.61803398875... which is (1+sqrt(5))/2.
Why not use x = x^2 -1?
Generally you try to reduce the degree of the polynomial you're trying to find the root for.
How did you pick x1?
Your starting point should be an educated guess, a point in the neighborhood of your root.
How can you use the convergence test without the root?
Think of the convergence test as more of "will this function converge to this root?" When you don't know the root, try iterating a few times to see if the function is converging, bouncing around in a loop, or going to infinity. It will become apparent very quickly.
What happens if a function fails the convergence test?
Failing the test means that the function is not guaranteed to converge. It might still converge but it makes no promises. Take the function which I showed fail in the example. If you iterate starting from the root that we found, the function might converge to the same value depending on your calculator's accuracy.
Doesn't this function have two roots? Is there a way to find the second one?
Indeed this function has two roots (1+sqrt(5))/2 and (1-sqrt(5))/2 which are the numbers φ (phi) and ψ (psi). I showed how the first example converged to phi and that the other did not for simplicity. You can use the second equation to converge on psi if you start close enough, like -1 for example.
Is there any way to use x = +/- sqrt(x + 1)?
In this case you can use x = sqrt(x+1) which will converge to 1.618 as long as the value inside the square root is positive. If you try to take the square root of a negative number you will have to use imaginary and complex numbers.
Is there a way to speed up Fixed Point Iteration?
Yes, check out my video on Steffensen's Method with Aitken's Δ² https://youtu.be/BTYTj0r5PZE and my video on Wegstein's Method https://youtu.be/T_6mR6rJXQQ
How can I force Fixed Point Iteration to converge?
There is a very simple change you can make to induce convergence called Wegstein's Method https://youtu.be/T_6mR6rJXQQ
Can you make a video that answers these questions?
Absolutely check out Fixed Point Iteration Q&A https://youtu.be/FyCviw2ZA2o
Chapters
0:00 Intro
0:06 Fixed Point Iteration
0:39 Fixed Point Iteration Example
2:12 Convergence Test
2:41 Convergence Test Example
3:18 Order
4:03 Thanks For Watching
Further Viewing:
Fixed Point Iteration Q&A https://youtu.be/FyCviw2ZA2o
Steffensen's Method with Aitken's Δ² https://youtu.be/BTYTj0r5PZE
Wegstein's Method https://youtu.be/T_6mR6rJXQQ
Fixed Point Iteration Systems of Equations https://youtu.be/xa2vUsYJD-c
Generalized Aitken-Steffensen Method https://youtu.be/-x9_fSNrX3w
#FixedPointIteration #NumericalAnalysis
Let’s talk about the fixed point iteration method, in particular the intuition behind the fixed point method. The fixed point method is an open-root finding method much like some other methods we have learned (Newton’s Method). We find are essentially setting up two functions that when equal will meet at a point in space directly above the root of our function.
This timeline is meant to help you better understand the Fixed Point Method Intuition:
0:00 Introduction.
0:05 What is the fixed-point method?
0:48 Fixed point method intuition
3:08 Outro
Relevant Numerical Methods Playlists:
Numerical Methods Playlist: https://www.youtube.com/watch?v=F2NWYEOHIkc&list=PLDea8VeK4MUTOBXLpvx_WKtVrMkojEh52
Numerical Methods Examples Playlist: https://www.youtube.com/watch?v=UA7bzwCwHMI&list=PLDea8VeK4MUTxZGOTG-VNnW4T3TYwlEc9
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This video is part of our Numerical Methods course. Numerical methods is about solving math problems through approximating the solution of problems that would be difficult or impossible to solve analytically. In this playlist we will cover topics such as solving systems of linear equations, solving systems of non-linear equations, numerical integration, numerical derivatives, etc..
#studysession #numericalmethods #fixedpoint
Floating point numbers are used a lot in computing from 3D graphics to the latest AI models, they are everywhere! I want to make a video about floating point numbers, but before that I think it is important to cover fixed point numbers. So this video is about fixed point numbers and the next one will be about floating point numbers!
---
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Twitter: https://twitter.com/garyexplains
Instagram: https://www.instagram.com/garyexplains/
#garyexplains
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
In this lesson, we shall consider the problem of finding the roots or solutions to fixed-point iteration systems.
This video covers the introduction to the topic. In the introduction, we shall solve three different examples. This video covers only Example 1.
00:00 - Example 1
Playlists on various Course
1. Applied Electricity
https://www.youtube.com/playlist?list=PLInywrvFyvq7pFsDEDu2-n0f5UOhpqWBD
2. Linear Algebra / Math 151
https://www.youtube.com/playlist?list=PLInywrvFyvq4IE-nW-ikwkZ2v81n31HQX
3. Basic Mechanics
https://www.youtube.com/playlist?list=PLInywrvFyvq6FUfAigJ3157kg-nZ020fd
4. Calculus with Analysis / Calculus 1 / Math 152
https://www.youtube.com/playlist?list=PLInywrvFyvq6_G3iA7LHbt5exJgGbp4Ok
5. Differential Equations / Math 251
https://www.youtube.com/playlist?list=PLInywrvFyvq408vWA5OYXShA6rlT51TdS
6. Electric Circuit Theory / Circuit Design
https://www.youtube.com/playlist?list=PLInywrvFyvq4sNicTbLBUpgkxrkcs2OGN
7. Calculus with Several Variables
https://www.youtube.com/playlist?list=PLInywrvFyvq5IXmWYBTTMvM_EdNcfUVt3
8. Numerical Analysis
https://www.youtube.com/playlist?list=PLInywrvFyvq7oAlPscVnXsd8CRTsh0b77
Make sure to watch till the end.
Like, share, and subscribe.
Thank you.
In this video, the difference between the Fixed Point and Floating Point Representation is explained. And how very large or very small numbers are stored in a computer using the floating point representation is explained. Also at the end, how to represent any Binary number in Floating Point representation is explained through examples.
Fixed Point Representation:
In Fixed Point Representation, the position of the radix point remains fixed. Because of the fixed radix point/ decimal point, for a given number of bits, the range of Binary numbers gets restricted.
Using 16 bits or 32 bits, in a fixed point representation, it is not possible to represent very large or very small numbers.
Floating Point Representation:
In Floating Point Representation, the position of the radix point / Binary Point is dynamic. Therefore, it is possible to represent very large or very small numbers.
The floating Point Number representation is similar to the scientific notation used for the decimal numbers. It is a normalized representation.
In Floating Point Representation, there is only a significant digit before the Binary point.
The floating Point Representation consists of 3 parts.
1) Sign
2) Significand
3) Exponent
In this video, how to represent any Binary Number in a floating point representation is explained using the examples.
Check out the other useful videos related to Digital Electronics (Playlist):
https://youtube.com/playlist?list=PLwjK_iyK4LLBC_so3odA64E2MLgIRKafl&si=wpYsuM0VFsvMwftG
The following topics are covered in the video:
0:00 Introduction
0:44 Fixed Point Numbers and their limitation
6:41 Floating Point Number Representation
This video will be helpful to all the students of science and engineering in understanding how very large or very small numbers are stored in a computer using the floating point number representation.
#allaboutelectronics
#digitalelectronics
#floatingpointnumbers
#numbersystem
Support the channel through membership program:
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The origin z^* = 0 is an attracting fixed point for the Newton function
N(z) = z - (exp(10 z^E) - 1)/(10 E z^(E-1) exp(10 z^E)). The attracting basin od z^* is the set of all points whose orbits converge to z^*. The black region is the exterior of the basin. The different colors depend on the speed of the convergence (in the red and yellow regions this speed is greater than in the regions violet or green).
In this video we will see the evolution of the attracting basin with the exponent E.
In this video I look at a simple Fixed Point Number implementation in C++. I use constexpr to let the IDE run the code without even compiling anything! The final result is a small template class that permits arbitrary precision fixed point calculations.
Patreon: https://www.patreon.com/javidx9
YouTube: https://www.youtube.com/javidx9
https://www.youtube.com/javidx9extra
Discord: https://discord.gg/WhwHUMV
Twitter: https://www.twitter.com/javidx9
Twitch: https://www.twitch.tv/javidx9
GitHub: https://www.github.com/onelonecoder
Homepage: https://www.onelonecoder.com
In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point (after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating-point number representation.
Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor has no floating point unit (FPU) or if fixed-point provides improved performance or accuracy for the application at hand. Most low-cost embeddedmicroprocessors and microcontrollers do not have an FPU.
Representation
A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type. For example, the value 1.23 can be represented as 1230 in a fixed-point data type with scaling factor of 1/1000, and the value 1,230,000 can be represented as 1230 with a scaling factor of 1000. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation.
Before we can talk about floating-point binary representation, we need to understand fixed-point notation. No worries, though. It's a short hop from unsigned binary integers to fixed-point notation.
Take aways: At the completion of this lesson, each student should be able to:
• identify powers of two correlating to bit positions to right of a binary point,
• convert from fixed-point binary to decimal, and
• convert from decimal with a fractional part to fixed-point binary.
#fixedpoint #binary #fraction
My video on Sesame Studios: https://www.youtube.com/watch?v=BTjAiyyG2sw
The Curiosity Box by Vsauce: https://www.curiositybox.com/
LINKS TO SOURCES BELOW!
My twitter: https://twitter.com/tweetsauce
My instagram: https://www.instagram.com/electricpants
DONG: https://www.youtube.com/dong
Moon Museum:
https://en.wikipedia.org/wiki/Moon_Museum
E.A.T. https://en.wikipedia.org/wiki/Experiments_in_Art_and_Technology
Brouwer’s fixed point theorem:
http://mathforum.org/mathimages/index.php/Brouwer_Fixed_Point_Theorem
https://simanaitissays.com/2015/01/19/combed-coconut-must-have-cowlicks/
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://www.youtube.com/watch?v=86uarqPBJ9s
Excellent interactive wolfram demo of Brouwer: http://demonstrations.wolfram.com/BrouwerFixedPointTheorem/
Online graphing calculator used was: https://www.desmos.com/
fixed points more generally:
http://www.math.uri.edu/~kulenm/mth381pr/fixedpoint/fixedpoint.html
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Fixed-point_iteration
Gullible
9s: http://mathforum.org/library/drmath/view/62561.html
Babylonian method:
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
Iterated equation solver you can use to run the Babylonian method: http://www.wolframalpha.com/widgets/view.jsp?id=e4a9b48a942efefa14b2773c31982c29
aleph fixed point:
https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/
http://cantorsattic.info/Aleph
My video on counting past infinity: https://www.youtube.com/watch?v=SrU9YDoXE88
map projections:
http://www.gis.osu.edu/misc/map-projections/
https://www.jasondavies.com/maps/transition/
antipodes: http://geographer-at-large.blogspot.co.uk/2011/10/where-is-your-antipode.html
antipode-finders:
https://www.jasondavies.com/maps/antipodes/
http://www.findlatitudeandlongitude.com/antipode-map/#.V-nBkZMrIUF
Pacific antipode image: http://i.imgur.com/dyJQ3EW.jpg
Borsuk-Ulam:
https://www.youtube.com/watch?v=5Px6fajpSio
https://www.youtube.com/watch?v=_sA3c0AhK34
https://www.youtube.com/watch?v=EhAs-vZDQG0
“four” is the cosmic number:
Standupmaths video about how four has four letters. So bummed I only found this AFTER I made this vid, check it out!! https://www.youtube.com/watch?v=LYKn0yUTIU4
http://theweeklyriddle.blogspot.com/2010/09/cosmic-number.html
“Idempotent”: https://en.wikipedia.org/wiki/Idempotence
protocol_7’s comment: https://www.reddit.com/r/math/comments/tw502/can_you_supply_an_elegant_proof/
Other topology pages:
http://www.solitaryroad.com/c775.html
https://en.wikipedia.org/wiki/Homeomorphism
wiki images:
https://commons.wikimedia.org/wiki/File:Robert_Rauschenberg_(1968).jpg
https://en.wikipedia.org/wiki/Claes_Oldenburg#/media/File:Claes_Oldenburg_(1970).jpg
Globe is from Ballerby & Co. Globemakers
http://www.bellerbyandco.com/
https://www.instagram.com/globemakers/
music from
http://www.youtube.com/JakeChudnow
and http://www.audionetwork.com
Any VFX that actually look cool are by https://www.youtube.com/c/ericlanglay -- I did the rest the best I could :)
How much does a floating point processor improve floating point operations? Dr Bagley installed one to find out - and explains how computers store the bits.
https://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: https://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
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Note: To find the fix points of a function means to find when the input values equal the output values. Since the equation y=x is purely a one-to-one function, to find any fixed points for any function, we set that function equal to x and solve.
Q1. Find the fixed points of f(x)=∛x
Q2. Find the fixed points of h(x)=x^2−3x
Q3. Find the fixed points of the second iterate of g(x)=x^2−1
A video explaining how fixed point maths works and why it is useful on CPUs that have no floating point units.
A written version of this is available on my website, linked below
Fixed point maths is a way to deal with decimal numbers, on systems where they don't have the ability to use decimals. The basic concept is to scale all numbers up by a certain amount to remove any decimal parts of the number. Then you can work with whole numbers only, scaling back down when needed.
What scale you use depends on the range of numbers needed for a particular situation. It can also depend on what the hardware is best at managing. A Z80 CPU for example can work with 16 bit values reasonably easily. And if you’re not trying to do complex accurate scientific calculations, the resolution of your machine’s screen is likely to be a factor as well.
But if you want a quick one to use, this explanation uses what is called 8.8 fixed numbers. That is a 16 bit value where the upper 8 bits are the whole number and the lower 8 bits the fractional part. With an imaginary binary point in the middle.
Link to website: https://docs.ncot.uk/computing-maths/fixed-point-arithmetic/index.html
Chapters
00:17 Maths is hard
00:56 Floating point maths
01:45 I have more videos!
03:53 What we're trying to do
04:57 Example with money
06:39 Introducing fixed point maths
08:28 negative numbers
10:39 Explaining fixed point
15:03 Summary
In this video we'll look at fixed point arithmetic. This is a technique for performing operations on numbers with fractional parts using integers, instead of the more common floating point.
Fixed point has some pretty amazing benefits, depending on how and when it is used. We can use bytes, short ints, ints, or long integers, we can select the precision and range of our variables.
Fixed point is very flexible and (sometimes) very fast. It's extremely good at certain types of tasks, for instance, working with image processing matrices. It can also do some things which floating point cannot, such as exactly represent 1/3 or 1/5.
Fixed point is not a hack, it's a fantastic tool to add to your belt if you've not yet come across it. If you have used fixed point in the past, I hope this tute is a good refresher!
Also, I forgot to mention something in the tute:
The technique I used to get the fractional part of the fixed point number literally gives us the bits that comprise the fractional part of the number. If your number is negative, then they will not be the correct bits if you print the fixed point to screen.
Facebook:
https://www.facebook.com/pages/WhatsaCreel/167732956665435
In this video, we look at the convergence of the method and its relation to the Fixed-point theorem. Please note there is a mistake at the end of the video 1/x-1 is not less than 1 and the iteration diverges. I will update the video.
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While all the computations could be executed with 32-bit fixed-point arithmetic, the most common and convenient format for mathematical calculations especially when dealing with long numbers has been for years the IEEE 754 floating-point standard.
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