Digital Revolution

Arithmetic in The Binary Number System

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451-0001

email: joeyc@cunyvm.cuny.edu
web page: www.york.cuny.edu/~malk

One one has mastered representing numbers in the binary system or base 2, it is necessary to learn how to carry out the arithmetic operations of addition, subtraction, multiplication and division of numbers in the binary system.

Luckily, we can easily transfer the ideas that you learned for doing arithmetic in decimal to doing arithmetic in binary. The major difference, after learning to get used to the process, is that arithmetic in binary is much easier than in decimal!

When you were in grade school you learned how to do problems such as:

123 + 236, 145 x 341, 453 - 249, and 87 divided by 16. Now you will have to deal with problems which in binary look like: 1101 + 1011, 1011 x 101, 1111- 1001, 10111 divided by 11.

To do decimal arithmetic you learned your addition and multiplication tables. These tables each had 100 entries. Typical addition facts were that 7 + 8 = 15 and 6 + 9 = 15 and typical multiplication facts were that 5 x 6 = 30 and 9 x 8 = 72.

What are the facts you need to do arithmetic in binary? Surprisingly, all you will need are 4 facts for addition and 4 facts for multiplication!

Here is the addition table in binary:

0 1

0 0 1

1 1 10

In words this table says that if you add 0 to any binary digit b (b can be 0 or 1) you get the value b, and that if you add the digit 1 to the digit 1 you get a 0 with a 1 "to carry." That is, in binary 1 + 1 is two but we write two as 10.

Here is the multiplication table in binary:

0 1

0 0 1

1 1 1

In words this table says that if you multiply any binary digit b (b can be 0 or 1) by 0 you get the value 0, and that if you multiply 1 by 1 you get 1.

As you see the addition and multiplication tables for binary arithmetic could not be any simpler, but like all new ideas one has to become accustomed how to use these tables.

The key to understanding binary is the analogy with decimal arithmetic:

342 = 3x(100) + 4(10) + 2 = 3(10)2 + 4(10)1 + 2(10)0

Hence, 1101 in the binary system means:

1011 = 1(8) + 0(4) + 1(2) + 1(1) = 1(2)3 + 0(2)2 + 1(2)1 + 1(2)0 = eleven = 11 in decimal.