Number bases: decimal (base 10)

The idea

Decimal is the base-10 number system you use every day. It has ten distinct digits, 0 to 9, and each position carries a place value that is a power of ten: the units column is 10⁰ = 1, the tens column is 10¹ = 10, the hundreds column is 10² = 100, and so on. The number 352 therefore means 3 × 100 + 5 × 10 + 2 × 1. Understanding decimal place value is the foundation for converting to and from binary and hexadecimal, because the same positional principle applies with a different base. You should be able to identify decimal as base 10, name its digits, and explain the value of each column.

What earns the marks

• Decimal is base 10.
• Decimal uses the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
• Each column has a place value that is a power of ten: 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000.
• The value of a decimal number is the sum of each digit multiplied by its column place value, for example 352 = 3 × 100 + 5 × 10 + 2 × 1.
• The same positional principle underlies other bases, so decimal place value helps explain binary and hexadecimal.

Common mistakes and how to put them right

• Treating the digits as having fixed values regardless of position.

  Put it right: In 352 the digit 3 is worth 300 because of its column, not 3.

• Confusing the base number with the number of columns.

  Put it right: Base 10 means ten digit symbols, not ten columns; columns extend indefinitely as powers of ten.

• Thinking decimal is the only valid number system.

  Put it right: Decimal is one base among several, and computers commonly use binary and hexadecimal.

Exam tips

• Write out the column headings (…, 1000, 100, 10, 1) before working through a place-value question.
• Remember that the rightmost column is always the units column with place value 1 (10⁰).
• Link decimal place value to binary and hexadecimal so the shared positional idea becomes clear.