Syllabus Numbers & Quantification Introduction to Binary Number System

Introduction to Binary Number System

A comprehensive guide to the Binary Number System for CBSE Class 11 Applied Mathematics, covering base-2 conversion, positional notation, and arithmetic operations.

Fundamentals of the Binary Number System

The Binary Number System is a positional numeral system that represents values using only two symbols: 0 and 1. Unlike the decimal system (base-10) which uses ten digits (0-9), the binary system is base-2. In this system, each position represents a power of 2, starting from the rightmost bit (2^0) and increasing as we move to the left.

Value = (d_n * 2^n) + (d_{n-1} * 2^{n-1}) + ... + (d_1 * 2^1) + (d_0 * 2^0), where d is either 0 or 1.
Example 1: Convert the binary number 1011_2 to its decimal equivalent.
Show Step-by-Step Solution

• Identify the positional values from right to left: (1 * 2^0) + (1 * 2^1) + (0 * 2^2) + (1 * 2^3)
• Calculate each term: (1 * 1) + (1 * 2) + (0 * 4) + (1 * 8)
• Sum the results: 1 + 2 + 0 + 8 = 11

Answer: The decimal equivalent of 1011_2 is 11.

Conversion from Decimal to Binary

To convert a decimal number to binary, perform repeated division by 2 and track the remainders. The binary representation is formed by reading the sequence of remainders in reverse order (from the last remainder to the first).

n_2 = (r_k, r_{k-1}, ..., r_1, r_0) where r is the remainder of successive division by 2.
Example 1: Convert the decimal number 13 to binary.
Show Step-by-Step Solution

• 13 ÷ 2 = 6, remainder 1
• 6 ÷ 2 = 3, remainder 0
• 3 ÷ 2 = 1, remainder 1
• 1 ÷ 2 = 0, remainder 1
• Read remainders from bottom to top: 1101

Answer: The binary equivalent of 13 is 1101_2.

Binary Addition Rules

Binary addition follows specific rules similar to decimal addition but is limited by the base-2 constraint. The rules are: 0+0=0; 0+1=1; 1+0=1; 1+1=10 (write 0, carry 1).

1 + 1 = 10_2
Example 1: Add 101_2 and 110_2.
Show Step-by-Step Solution

• Align digits: 101 + 110
• Rightmost column: 1 + 0 = 1
• Middle column: 0 + 1 = 1
• Leftmost column: 1 + 1 = 10
• Combine results: 1011

Answer: 101_2 + 110_2 = 1011_2.