1.1b Binary System

Binary System

A binary number is a number expressed in the base-2 numeral system, which uses only two symbols: 0 and 1. Each digit in a binary number is called a bit. $N$ bits can represent exactly $2^N$ numbers. Binary numbers are often prefixed with 0b to distinguish them from numbers in other bases.

To convert a number $x_2$ to base-$2^k$:

1. Group the bits into clusters of $k$ bits.
2. Convert each bit to base-10, and then to base $2^k$.

To reverse the conversion:

1. Convert each digit of the base-$2^k$ number to decimal.
2. Convert to binary and cluster each digit into bit clusters of size $k$.

To convert a number from and to base-$n$:

1. Convert the number to base-10 and follow the steps for conversion.

Intuition

The following is an easy table that shows you how to convert from hexadecimal to binary: