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Digital Computers and Related Mathematics MAA
Digital Computers and Related Mathematics
Binary Numbers

Let us see how it is possible to Write the proper equations to make a com-
puter add two numbers. Before doing so, we must look at the number system
actually used internally by computers. Although we use a number system
based on 10, the decimal system, computers usually translate these numbers to
base 2, the binary system, for internal operation. In the decimal system, 3257
means 3- 103 + 2- 102 + 5- 10 + 7. In the binary system, 1101 means 1 - 23 +
1- 22+0 - 2 +1. The desirable property of the binary system is that only
two symbols are needed, 0 and 1. Let us look at the binary representation of some numbers.

DECIMAL BINARY 

0 0        9 1001
1 1     10 1010
2 10   11 1011
3 11   12 1100
4 100 13 1101
5 101 14 1110
6 110 15 1111
7 111 16 10000
8 1000

Table 1
Examples.

1. The decimal number 15, when expressed in terms of powers of 2, is
1'23+1'22+1'2+1, 0r8+4+2+ 1. Thus, its binary form is 1111.

2. The decimal number .125 is 1/8, which is (1/2)3, or 2-3. Since 2-3 = 0- 2-1 +0 ' 2-2 -1- 1 - 2-3, its binary form is .001.

3. The binary number .101 represents 1 - 2'1 + 0- 2-2 +1 - 2-3, or 1(1/2)+
0(1/4) + 1(1/8) = 1/2 + 1/8 = .5+ .125 = .625.

Problem 1. Verify the entries in the following table.

DECIMAL BINARY
25 = 11001
31 = 11111
32 = 100000

.5 = .1
.75 = .11
.25 = .01

Registers

Each flip flop can store one binary digit (usually abbreviated to bit). A
sequence of flip-flops, then, can store a number in binary form. Such a se-
quence is called a register. Let us suppose that we have two registers, each
consisting of 4 flip-flops. We shall call one of these the A-register, and labelthe flip-flops, from left to right, A1, A2, A3, and A4. The other register will be called the B-register, and will consist of flip-flops B1, B2, B3, and B4.

A1 A2 A3 A4 B1 B2 B3 B4
 
 
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