Introduction to the Binary Number System

In a previous blog post, I discussed the decimal number system, which is the most commonly used system in human history. However, when it comes to electronics and computers, the binary number system takes center stage.

In electronics, we have only two states: “on” or “off”, represented by 1 and 0 respectively. This simple yet powerful concept forms the foundation of computer systems. Each digit in the binary number system is called a bit.

Similar to the decimal system, the binary system is also positional. We calculate the value of each digit by multiplying it by a power of 2, depending on its position. Starting from the right side, the first position is 0, followed by 1, 2, 3, and so on. The values of the powers of 2 are as follows:

  • 2^0 equals 1
  • 2^1 equals 2
  • 2^2 equals 4
  • 2^3 equals 8
  • and so on…

Using this positional system, we can represent numbers using a series of bits. For example:

  • The number 1 can be represented as 1 x 2^0
  • The number 10 can be represented as 1 x 2^1 + 0 x 2^0
  • The number 111 can be represented as 1 x 2^2 + 1 x 2^1 + 1 x 2^0

Leading zeros in a binary number are insignificant and can be dropped or added as needed. For example, 110 can be represented as 0110 or 00000110. It’s important to note that leading zeros on the left of the first 1 do not change the value because multiplying any power of 2 by 0 is always 0.

With binary numbers, we can represent any decimal number. The number of bits determines the range of numbers that can be represented. For example, with 4 bits (bits are also referred to as binary digits), we can represent numbers from 0 to 15 (2^4 - 1). In general, the number of possible numbers that can be represented with n bits is 2^n.

Let’s take a look at some simple conversion tables:

Conversion table for the first 4 digits (2 bits):

Decimal number Binary number
0 00
1 01
2 10
3 11

Conversion table for the first 8 digits (3 bits):

Decimal number Binary number
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111

As you can see, the patterns repeat as we increase the number of bits. We simply prepend 0 to the first set of numbers (0-7) and prepend 1 to the second set of numbers (8-15).

In future blog posts, I will discuss how to perform operations such as addition and division with binary numbers, explore the hexadecimal number system, and explain how to convert between binary, decimal, and hexadecimal without relying on conversion tables.

Tags: binary number system, computer science, positional number system, bits, conversion tables.