binary multiplication. any multiplication can be re-expressed as a series of additions. for example,...
TRANSCRIPT
![Page 1: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/1.jpg)
Binary Multiplication
![Page 2: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/2.jpg)
Binary Multiplication
Any multiplication can be re-expressed as a series of additions.
For example, 7 * 3 (seven times three) is simply the sum of 3 sevens.
7 * 3 = 7 + 7 + 7Since we already know how to perform
addition, binary multiplication becomes easy.
![Page 3: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/3.jpg)
Binary Multiplication
7
* 3
![Page 4: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/4.jpg)
Binary Multiplication
7 0111
* 3
![Page 5: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/5.jpg)
Binary Multiplication
7 0111
* 3 + 0111
+ 0111
![Page 6: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/6.jpg)
Binary Multiplication
7 0111
* 3 + 0111
1110
+ 0111
![Page 7: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/7.jpg)
Binary Multiplication
7 0111
* 3 + 0111
1110
+ 0111
10101
![Page 8: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/8.jpg)
Binary Multiplication
The multiplication of large values will take many such additions, but the machine doesn’t mind.
Even so, we look for faster more efficient techniques, and there is one available.
It’s based on the Algebraic property called Distribution.
![Page 9: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/9.jpg)
Binary Multiplication
The Distribution Property can be expressed this way:
a * (x + y) = a * x + a * yAny number can be expressed as the sum of
other numbers:106 * 37 = 106 * (30 + 7) = (106 * 30) + (106 * 7)
![Page 10: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/10.jpg)
Binary Multiplication
The other insight necessary to optimize multiplication regards multiplying a number by its base.
In Decimal, multiplying anything by 10 is easy - simply shift the digits to the left, and pad with a zero.
975 * 10 = 9750
![Page 11: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/11.jpg)
Binary Multiplication
This is true in any number system.
In Binary, for example:
001 represents 1
010 represents 2
100 represents 4
![Page 12: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/12.jpg)
Binary Multiplication
This is true in any number system.
In Binary, for example:
001 represents 1
010 represents 2
100 represents 4
Each shift/pad doubles the value.
![Page 13: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/13.jpg)
Binary Multiplication
Now reconsider this problem:
106 * 37 = ?
![Page 14: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/14.jpg)
Binary Multiplication
Now reconsider this problem:
106 * 37 = ?Expressed as this distribution:
106 * (32 + 4 + 1)
![Page 15: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/15.jpg)
Binary Multiplication
Now reconsider this problem:
106 * 37 = ?Expressed as this distribution:
106 * (32 + 4 + 1)
= 106 * 32 + 106 * 4 + 106 * 1
![Page 16: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/16.jpg)
Binary Multiplication
? = 106 * 32 + 106 * 4 + 106 * 1
![Page 17: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/17.jpg)
Binary Multiplication
? = 106 * 32 + 106 * 4 + 106 * 1
32 = (25)
![Page 18: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/18.jpg)
Binary Multiplication
? = 106 * 32 + 106 * 4 + 106 * 1
32 = (25)
4 = (22)
![Page 19: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/19.jpg)
Binary Multiplication
? = 106 * 32 + 106 * 4 + 106 * 1
32 = (25)
4 = (22)
1 = (20)
![Page 20: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/20.jpg)
Binary Multiplication
? = 106 * 32 + 106 * 4 + 106 * 1
32 = (25)
4 = (22)
1 = (20)by substitution…
? = 106 * 25+ 106 * 22 + 106 * 20
![Page 21: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/21.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
This term can be
simplified.
![Page 22: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/22.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
106
![Page 23: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/23.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
106 converts to 1101010
![Page 24: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/24.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
1101010
106 * 22
![Page 25: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/25.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
1101010
106 * 22 shift 2
![Page 26: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/26.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
1101010
106 * 22 shift 2 110101000
![Page 27: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/27.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
1101010
110101000
106 * 25
![Page 28: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/28.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
1101010
110101000
106 * 25 shift 5
![Page 29: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/29.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
1101010
110101000
106 * 25 shift 5 110101000000
![Page 30: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/30.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
106 1101010
106 * 22 110101000
106 * 25 110101000000====================================
![Page 31: Binary Multiplication. Any multiplication can be re-expressed as a series of additions. For example, 7 * 3 (seven times three) is simply the sum of 3](https://reader030.vdocument.in/reader030/viewer/2022033106/56649d845503460f94a6aab6/html5/thumbnails/31.jpg)
Binary Multiplication
? = 106 * 25+ 106 * 22 + 106 * 20
106 1101010
106 * 22 110101000
106 * 25 110101000000====================================
Sum 111101010010