Note #5
An Introduction to Binary Notation

Binary - a number system used to encode information using 0s and 1s - lies at the heart of every digital computer.

What is binary?

Binary is a base-2 number system, which only uses two digits (0 & 1). It is a system used at the heart of every digital computer, allowing them to encode information, perform arithmetic operations and execute logical control processes.

Using two digits as opposed to, say, the familiar ten digits used in decimal systems (0 to 9) enables hardware to be easily implemented via simple ‘on’ or ‘off’ circuit states or logic gates. This is the basis for all digital systems.

Understanding binary notation

To understand binary values imagine each digit (or ‘bit’) of the binary notation as representing an increasing power of 2 - with the rightmost digit representing 20, the next representing 21, then 22 and so on.

2n 23 22 21 20
Decimal 8 4 2 1
Figure 1. Increasing powers of two represented with its decimal result.

As a visual example, figure 2 shows the binary value 1100 converted into the decimal value 12. Blue numbers represent the binary notation, red numbers represent the increasing powers of 2, and green numbers represent the decimal values.

Representation of 12 in binary.
Figure 2. Representation of 12 in binary.

Binary values are often represented in different ‘bit’ lengths or ‘word’ sizes. In the example above the value is represented in 4-bits, called a ‘nibble’. This means a value between 0 and 15 can be represented. 8-bit values, referred to as ‘bytes’ can represent a value between 0 and 255. Figure 3 shows the 4-bit binary representation for each decimal value between 0 and 15.

Decimal Binary
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
Figure 3. Binary table of 4-bit values from 0 to 15.

Learn more with the ARITH-MATIC S1-AU Mk1

If you'd like to find out more about binary in computational machines, or how to perform binary arithmetic in a visual and tactile way, check out the S1-AU Mk1: a 4-bit arithmetic unit designed to beautifully dissect the complex mechanisms of computation.

Note by ARITH-MATIC
November 2018

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