Place Value Concepts

Each position has a value

In our decimal number system, a digit's position tells you its value. Moving one place left makes that place worth ten times as much; moving one place right makes it worth one tenth as much. In 4,376, the 3 means three hundreds, or 300. The decimal point separates the ones from the tenths: in 4,376.052, the 5 means five hundredths, or 0.05.

Zero can hold a place even when there is nothing of that value. In 4,376.052, the 0 shows there are no tenths; without it, the 5 would be in the tenths place. The same idea lets us write very large and very small numbers: 5,000,000 is five million, while 0.000005 is five millionths. Read from the decimal point to identify a digit's place rather than guessing from the number of zeros.

Scaling by powers of ten

Multiplying by 10 moves each digit one place to the left in the place-value chart, so 3.46 × 10 = 34.6. Multiplying by 100 moves each digit two places left, so 3.46 × 100 = 346. Dividing by 10 or 100 moves the digits one or two places right instead. Zeros fill any empty places: 52 ÷ 1,000 = 0.052.

This also explains calculations with numbers between 0 and 1. Multiplying by 0.1 is the same as dividing by 10, so 38 × 0.1 = 3.8. Dividing by 0.1 has the opposite effect: 38 ÷ 0.1 = 380. Dividing by a number less than 1 can therefore give an answer larger than the starting number.

Keeping the places correct in decimal calculations

For addition and subtraction, combine digits with matching place values. Write 2.7 as 2.70 when adding 0.36: 2.70 + 0.36 = 3.06. The extra zero changes the way the number is written, not its value.

For multiplication, the place values determine the size of the answer. Since 6.34 = 634 ÷ 100 and 5.2 = 52 ÷ 10, calculate 634 × 52 = 32,968, then divide by 1,000. Therefore 6.34 × 5.2 = 32.968. An estimate, 6 × 5 ≈ 30, confirms that 32.968 is a sensible size.

For division by a decimal, scale both numbers by the same power of ten to make the divisor a whole number. For example, 4.8 ÷ 0.06 = 480 ÷ 6 = 80. Multiplying both numbers by 100 leaves the quotient unchanged.

Place value also lets you reuse a known calculation. If 24 × 35 = 840, then 2.4 × 3.5 = 8.4: each factor is one tenth as large, so the product is one hundredth of 840.