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Binary Math

Binary numbers and mathematics.

Addition & Subtraction


Let's first take a look at decimal addition.

As an example we have 26 plus 36,

    26
  +36

To add these two numbers, we first consider the "ones" column and calculate 6 plus 6, which results in 12. Since 12 is greater than 9 (remembering that base 10 operates with digits 0-9), we "carry" the 1 from the "ones" column to the "tens column" and leave the 2 in the "ones" column.

Considering the "tens" column, we calculate 1 + (2 + 3), which results in 6. Since 6 is less than 9, there is nothing to "carry" and we leave 6 in the "tens" column.

    26
  +36
    62


Binary addition

  works in the same way, except that only 0's and 1's can be used, instead of the whole spectrum of 0-9. This actually makes binary addition much simpler than decimal addition, as we only need to remember the following:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10

As an example of binary addition we have,

  101
+101

a) To add these two numbers, we first consider the "ones" column and calculate 1 + 1, which (in binary) results in 10. We "carry" the 1 to the "tens" column, and the leave the 0 in the "ones" column.

b) Moving on to the "tens" column, we calculate 1 + (0 + 0), which gives 1. Nothing "carries" to the "hundreds" column, and we leave the 1 in the "tens" column.

c) Moving on to the "hundreds" column, we calculate 1 + 1, which gives 10. We "carry" the 1 to the "thousands" column, leaving the 0 in the "hundreds" column.

  101
+101
1010

Another example of binary addition:

  1011
+1011
10110

Note that in the "tens" column, we have 1 + (1 + 1), where the first 1 is "carried" from the "ones" column. Recall that in binary,

1 + 1 + 1 = 10 + 1
                = 11


Binary subtraction

  is simplified as well, as long as we remember how subtraction and the base 2 number system. Let's first look at an easy example.

  111
 - 10
  101

Note that the difference is the same if this was decimal subtraction. Also similar to decimal subtraction is the concept of "borrowing." Watch as "borrowing" occurs when a larger digit, say 8, is subtracted from a smaller digit, say 5, as shown below in decimal subtraction.

    35
  -   8
    27

For 10 minus 1, 1 is borrowed from the "tens" column for use in the "ones" column, leaving the "tens" column with only 2. The following examples show "borrowing" in binary subtraction.

    10       100      1010
   -  1       - 10     - 110
      1          10       100


To practice binary addition and subtraction, visit the Practice Exercises page.

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Binary Numbers | Add & Subtract | Multiply & Divide | Exercises | Conversion