Computer Science – 1.1 Data Representation | e-Consult

Binary magnitudes represent the power of 2 associated with each position in a binary number. Unlike decimal magnitudes which are powers of 10 (1, 10, 100, 1000, etc.), binary magnitudes are powers of 2 (1, 2, 4, 8, 16, etc.). Each position in a binary number corresponds to a power of 2, starting from the rightmost digit as 2⁰, then 2¹, 2², and so on.

For example, consider the binary number 1011₂. The magnitudes are:

• Rightmost digit (1): 2⁰ = 1
• Second digit from right (1): 2¹ = 2
• Third digit from right (0): 2² = 4
• Leftmost digit (1): 2³ = 8

The decimal equivalent of 1011₂ is (1 * 8) + (0 * 4) + (1 * 2) + (1 * 1) = 8 + 0 + 2 + 1 = 11.

The difference is crucial because in decimal, each position represents a multiple of 10. In binary, each position represents a multiple of 2. This means that a value in binary that requires a relatively small number of digits in decimal will require significantly more digits in binary. This is because the powers of 2 grow much faster than the powers of 10.

Consider the number 100 (decimal). This is represented as 1100100 (binary). This requires 7 digits. The same number, 100 (decimal), is represented as 100 (binary), which only requires 3 digits. This illustrates the difference in magnitude representation.