A Story of Numbers

Chapter focus

A numeral is a way of recording a number. Comparing how different notations express quantity shows why place value, a zero symbol and a consistent base make calculations efficient.

These original explanations and practice prompts support revision. Follow the full activity and discussion sequence in your assigned NCERT edition.

1. Numbers, numerals and symbols

A number is the quantity or value; a numeral is the written symbol used to represent it. The value five, for example, can be written as 5, V or five in words. Different numeral systems use different symbols and rules while representing quantities.

A positional system assigns a value to a symbol according to its position. In our decimal system, each place to the left is ten times the place before it. A compact set of digits can represent many values by changing positions.

2. Place value and the role of zero

In 4,206, the digit 4 means four thousands, 2 means two hundreds, 0 means no tens, and 6 means six ones:

4,206 = 4 × 1,000 + 2 × 100 + 0 × 10 + 6 × 1.

Zero has two connected uses: it can represent the quantity none, and it can hold an empty place in a numeral. The zero in 205 tells us there are no tens. Without it, 25 would mean two tens and five ones—a different value.

Compare 3,407 and 347. In 3,407, 3 represents three thousands; in 347, it represents three hundreds. The digit is the same, but its position changes its value.

3. The base tells the place values

In base b, the places from right to left are 1, b, b², b³, and so on. A base-b numeral expands as the sum of each digit multiplied by its place value. Permitted digits are 0 through b−1.

Base four uses 0, 1, 2 and 3. Its places are 1, 4, 16, 64, … . The numeral 132₄ means:

1 × 16 + 3 × 4 + 2 × 1 = 16 + 12 + 2 = 30 in base ten.

The subscript identifies the base. In base four, the digit 4 is not allowed because each digit must be less than the base.

4. Convert from another base to decimal

Worked example: Convert 231₄ to base ten

1. Write the place values: 4², 4¹, 4⁰ = 16, 4, 1.
2. Multiply each digit by its place: 2 × 16, 3 × 4, 1 × 1.
3. Add: 32 + 12 + 1 = 45. Thus 231₄ = 45₁₀.

Check that every digit is less than 4 and reverse the expansion: 2 × 16 + 3 × 4 + 1 = 45.

5. Convert decimal numbers to another base

Repeated division by the new base gives digits from right to left. Record each remainder, then read them upwards from the last division.

Worked example: Write 45 in base four

1. 45 ÷ 4 = 11 remainder 1.
2. 11 ÷ 4 = 2 remainder 3.
3. 2 ÷ 4 = 0 remainder 2.
4. Read upwards: 231₄. Therefore 45₁₀ = 231₄.

Reverse-check: 2 × 16 + 3 × 4 + 1 = 45. This catches digit-order errors.

6. Why positional notation is useful

Place value supports short written methods because each column has a known value. When a column reaches the base, regroup into one unit of the next place. In decimal addition, ten ones regroup as one ten; in base four, four ones regroup as one four.

The same idea works in any positional base, although the digits and regrouping threshold change. To compare numerals in the same base, compare their number of places first; if these match, compare digits from left to right.

Common errors to avoid

• Do not treat a base-four numeral as decimal without converting it.
• Do not use a digit equal to or greater than the base.
• When converting by repeated division, read remainders from last to first.
• Zero may mark an empty position and affect the value of the whole numeral.

Quick revision

Practice: show each step

Chapter summary

Numeral systems encode numbers using symbols and rules. Place-value systems assign powers of a base to positions, while zero can represent nothing and preserve an empty place. Expand a numeral to convert it to decimal; use repeated division and remainders to convert decimal numbers to another base.

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