Positional or Weighted Number System

The traditional number systems that we learned in school and use every day (decimal, binary, hexadecimal, octal etc.) are positional number systems. In such a system, a number is represented by a string of digits where each digit position has an associated weight. The value of a number is a weighted sum of the digits.

• Example 1: 1734 = (1 x 1000) + (7 x 100) + (3 x 10) + (4 x 1)
• Example 2: 568.23 = (5 x 100) + (6 x 10) + (8 x 1) + (2 x 0.1) + (3 x 0.01)

In general, a number D of the form d₁d₀.d_-1d_-2 has the value

D = d₁ x 10¹+ d₀ x 10⁰ + d_-1 x 10^-1 + d_-2 x 10^-2

Here 10 is called the radix or base.

Few Relevant Questions and Answers

Q1. Express the decimal number 106.58 as a sum of the values of each digit.

Answer: 106.58 = 1×10²+0x10¹+6×10⁰+5×10^-1+8×10^-2=1×100+0x10+6×1+5×0.1+8×0.01

Q2. What weight does the digit 7 have in each of the following numbers?

1. 1370 – Ans. 10
2. 6725 – Ans. 100
3. 7051 – Ans. 1000
4. 58.72 – Ans. 0.1

Weighting Structure of Other Number Systems

• Binary: 1011.01 = 1×2³ + 0x2² + 1×2¹ + 1×2⁰ + 1×2^-1 + 1×2^-2
• Hexadecimal: F1A2 = Fx16³ + 1×16² + Ax16¹ + 2×16⁰
• Octal: 436.7 = 4×8² + 3×8¹ + 6×8⁰ + 7×8^-1
• Base-4: 233 = 2×4² + 3×4¹ + 3×4⁰