Decimal numbers and place value

What it is

Our number system is a place-value system with base ten: there are ten digits, 0–9, and a digit’s value depends on the place it stands in. Each step to the left makes the place worth ten times as much, and each step to the right ten times less. The places are called ones, tens, hundreds, thousands, … to the left, and tenths, hundredths, thousandths, … to the right of the decimal point.

• 305.7 = 3 · 100 + 0 · 10 + 5 · 1 + 7 · 0.1. The 3 is worth 300, the 0 is worth no tens, the 5 is worth 5 and the 7 is worth 7 tenths. The 0 is there to hold the places: without it the number would be 35.7.
• 3,405: the 3 is worth 3,000, the 4 is worth 400, the 0 no tens and the 5 is worth 5.
• 4.07 = 4 + 0 tenths + 7 hundredths.
• 0.5 = 0.50: zeros at the end of the decimals do not change the value.
• 4.07 · 10 = 40.7: multiplying by 10 moves every digit one place to the left. Dividing by 10 moves it one place to the right.

A decimal number is a number written with a decimal point: the places to the right of the point continue the same system. It is another way to write fractions with denominator 10, 100, 1,000, …: 0.75 = 75/100 = 3/4.

Non-examples:

• 0.05 is not the same number as 0.5. The digit 5 stands in different places: hundredths against tenths.
• 305 is not 35. The zero is a placeholder.

Why it matters

The place-value system lets whole numbers and decimals be written with just ten symbols, and lets calculation be done in writing, column by column. Without it, adding large numbers would be hard, as it is with Roman numerals.

• Rounding means choosing which place to stop at, see rounding and significant figures.
• Scientific notation is place value with the powers of ten written out: 3,050 = 3.05 · 10³, see scientific notation.
• Fractions and decimals are two ways to write the same number, see fraction to decimal: 1/8 = 0.125.
• Percent is hundredths: 0.25 = 25%.

The place value worksheets practise the digit in a place, the value of a digit and expanded form.

Common misconceptions

• “0.25 is larger than 0.3, because 25 is larger than 3.” Compare place by place: 0.25 = 0.25 and 0.3 = 0.30, and 0.30 is larger. Pad with zeros until both have the same number of decimals.
• “After the point come whole numbers.” 3.14 is not 3 and 14: the 14 means 1 tenth and 4 hundredths. So 3.5 is larger than 3.14 even though 5 is less than 14.
• “A zero at the end matters.” 2.50 = 2.5. But the zero in 2.05 is a placeholder and cannot be dropped.
• “Multiplying by 10 means adding a zero.” That works for whole numbers (7 · 10 = 70) but not for decimals: 4.5 · 10 = 45, not 4.50.
• “A digit’s value equals the digit.” In 305.7 the 5 is worth 5 but the 3 is worth 300.

Common questions