https://cpudev.org/w/index.php?action=history&feed=atom&title=Binary_arithmeticBinary arithmetic - Revision history2026-04-25T16:37:49ZRevision history for this page on the wikiMediaWiki 1.36.1https://cpudev.org/w/index.php?title=Binary_arithmetic&diff=16&oldid=prevDemindiro: First draft. Written under the assumption that readers are not familiar with binary arithmetic yet, so heavy on examples.2023-02-18T13:32:04Z<p>First draft. Written under the assumption that readers are not familiar with binary arithmetic yet, so heavy on examples.</p>
<p><b>New page</b></p><div>As the name digital implies, digital computers work on on bits with a value of either 0 or 1.<br />
Accordingly, all operations, including arithmetic, operate on 0 or 1 values.<br />
<br />
Numbers that consists only of 0 or 1s are called '''binary''' or '''base2''' numbers.<br />
Binary numbers are often suffixed with a subscript 2, for example 1011011<sub>2</sub>.<br />
<br />
= Converting between decimal and binary =<br />
<br />
== Decimal as a sum of powers of 10 ==<br />
<br />
Any decimal number can be written as a sum of power of 10s.<br />
For example:<br />
<br />
* 42 = 4*10<sup>1</sup> + 4*10<sup>0</sup><br />
* 57005 = 5*10<sup>4</sup> + 7*10<sup>3</sup> + 0*10<sup>2</sup> + 0*10<sup>1</sup> + 5*10<sup>0</sup><br />
* 48.879 = 4*10<sup>1</sup> + 8*10<sup>0</sup> + 8*10<sup>-1</sup> + 7*10<sup>-2</sup> + 9*10<sup>-3</sup><br />
<br />
Binary numbers can be written in the same way though with powers of 2 instead of 10.<br />
<br />
== Converting from binary to decimal ==<br />
<br />
* 1010<sub>2</sub> = 1*2<sup>3</sup> + 0*2<sup>2</sup> + 1*2<sup>1</sup> + 0*2<sup>0</sup> = 8 + 2 = 10<br />
* 100.01<sub>2</sub> = 2<sup>2</sup> + 2<sup>-2</sup> = 4 + 1/4 = 4.25<br />
<br />
== Converting from decimal to binary ==<br />
<br />
# Divide the decimal number by 2.<br />
# Append the remainder to the ''left'' of the binary number.<br />
# Repeat with quotient as dividend until it is 0.<br />
<br />
For example:<br />
<br />
* 29 = 11101<sub>2</sub><br />
** 29 / 2 = 14, remainder 1<br />
** 14 / 2 = 7, remainder 0<br />
** 7 / 2 = 3, remainder 1<br />
** 3 / 2 = 1, remainder 1<br />
** 1 / 2 = 0, remainder 1<br />
<br />
The same technique can be used for rational numbers.<br />
<br />
# First multiply by some power of 2 to convert it to an integer.<br />
# Apply above procedure.<br />
# Divide by the same power of 2 to get the correct result.<br />
<br />
= Mathematical operations =<br />
<br />
The same techniques used for addition, multiplication ... used for decimal numbers can also be used for binary numbers.<br />
<br />
== Addition & subtraction ==<br />
<br />
22 10110<sub>2</sub><br />
+ 19 + 10011<sub>2</sub><br />
---- --------<br />
41 101001<sub>2</sub><br />
<br />
== Multiplication ==<br />
<br />
12 1100<sub>2</sub><br />
* 5 * 101<sub>2</sub><br />
---- -------<br />
10 1100<sub>2</sub><br />
+ 50 0<sub>2</sub><br />
---- + 110000<sub>2</sub><br />
60 ---------<br />
111100<sub>2</sub><br />
<br />
== Division ==<br />
<br />
372 | 5 101110100 | 101<sub>2</sub><br />
- 35 +---- - 101 +---------<br />
---- | 74 ----- | 1001010<sub>2</sub><br />
22 | 01 |<br />
- 20 | - 0 |<br />
---- | ----- |<br />
2 | 11 |<br />
- 0 |<br />
----- |<br />
110 |<br />
- 101 |<br />
----- |<br />
11 |<br />
- 0 |<br />
----- |<br />
110 |<br />
- 101 |<br />
----- |<br />
10 |<br />
- 0 |<br />
----- |<br />
10 |</div>Demindiro