Bitwise math/Introduction
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Binary math and hexadecimal math have slight differences to decimal math but the same principles apply. For example, in the decimal number 1234, the ‘4’ is in the “ones” placeholder, the ‘3’ is in the “tens” placeholder, the ‘2’ in the “hundreds” placeholder and the ‘1’ in the “thousands” placeholder.
Example 1: Decimal number 1234
| Thousands (1x10^3) | Hundreds (1x10^2) | Tens (1x10^1) | Ones (1x10^0) |
|---|---|---|---|
| 1 | 2 | 3 | 4 |
Binary operates a little bit differently, instead of having 1’s, 10’s, 100’s, etc, it has 1’s, 2’s, 4’s, 8’s, etc (That is to say, decimal math operates on a base of 10, while binary math operates on a base of 2). So analysing this for a moment, the binary number 1010, has a ‘1’ in the “eights” placeholder and a ‘1’ in the “twos” placeholder. Add these together and 10 is obtained in decimal numbers.
| Eights (1x2^3) | Fours (1x2^2) | Twos (1x2^1) | Ones (1x2^0) |
|---|---|---|---|
| 1 | 0 | 1 | 0 |
Another example is 1111:
| Eights (1x2^3) | Fours (1x2^2) | Twos (1x2^1) | Ones (1x2^0) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| Eights (1x2^3) | Fours (1x2^2) | Twos (1x2^1) | Ones (1x2^0) | |
|---|---|---|---|---|
| Binary Values | 1 | 1 | 1 | 1 |
| Decimal Values | 8 | 4 | 2 | 1 |
Through the use of the binary table, the binary values can each be multiplied (1’s) by the above multipliers (1x2^3 1x2^2 1x2^1 1x2^0) or “8, 4, 2, 1”. This will then give us, 8+4+2+1=15.
Basic Addition
Much like the decimal system, binary numbers can be added together. For this example the binary numbers 0110 and 0010 are going to be used and added.
| Eights | Fours | Twos | Ones | Total | |
|---|---|---|---|---|---|
| Binary | 0 | 1 | 1 | 0 | 6 |
| Decimal | 0 | 4 | 2 | 0 | 6 |
| Eights | Fours | Twos | Ones | Total | |
|---|---|---|---|---|---|
| Binary | 0 | 0 | 1 | 0 | 2 |
| Decimal | 0 | 0 | 2 | 0 | 2 |
Thus by using decimal addition, 4+2+2=8, the value of the addition of two binary numbers can be determined as 8.
