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Playing with Numbers: Binary

Binary (Base-2)

The binary system is a foundational concept in computing, representing numeric values using only two symbols: 0 and 1.

Digits

The only digits used are 0 and 1.

Converting Binary to Decimal

To convert a binary number to its decimal equivalent, you multiply each binary digit by 2 raised to the power of its position (starting from 0 on the right) and sum the results.

Example: Convert 10011 in binary to decimal.

The positions are 4, 3, 2, 1, 0 from left to right.

(1 * 2⁴) + (0 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰)
= (1 * 16) + (0 * 8) + (0 * 4) + (1 * 2) + (1 * 1)
= 16 + 0 + 0 + 2 + 1
= 19

Converting Decimal to Binary

To convert a decimal number to binary, you repeatedly divide the number by 2, keeping track of the remainders. The binary representation is the sequence of remainders read in reverse order.

Example: Convert 6 in decimal to binary.

  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1

Reading the remainders in reverse order gives 110. So, 6 in decimal is 110 in binary.

Binary Addition

Binary addition follows these simple rules for each bit:

  • 0 + 0 = 0 (carry 0)
  • 0 + 1 = 1 (carry 0)
  • 1 + 0 = 1 (carry 0)
  • 1 + 1 = 0 (carry 1)

Two’s Complement Representation

This is a method to represent both positive and negative numbers in binary.

  • Sign Bit: The leftmost bit (Most Significant Bit) indicates the sign. 0 for positive, 1 for negative.
  • Range: With n bits, you can represent numbers from -2ⁿ⁻¹ to 2ⁿ⁻¹ - 1.
  • Negating a Number:
    1. Flip all the bits (0s become 1s and vice versa).
    2. Add 1 to the result.

Example: Find the 4-bit Two's Complement for -4.

  1. Start with 4 in decimal which is 0100 in binary.
  2. Flip the bits: 1011.
  3. Add 1: 1011 + 1 = 1100.

So, -4 in decimal is represented as 1100 in 4-bit Two's Complement.

Converting from Two's Complement: If the sign bit is 0, convert as usual. If it's 1, it's a negative number. To find its value, negate it again (flip bits and add 1), convert the result to decimal, and put a negative sign in front.

Read more on the Wikipedia page for Binary numbers.