Demystifying - Multiplying and dividing by powers of ten

Introduction

Before this page: know the place-value columns on either side of the decimal point, and that each is worth ten times the column to its right.

“Add a zero” is the first rule most people learn for multiplying by ten, and it is the first one to break. It works on 52 and it fails on 5.2, and because it fails silently it is usually trusted long after it has stopped being true.

What is actually happening is simpler, and it never changes. The digits of a number keep their identity and their order; only their places change. Multiplying by 10 moves every digit one place to the left, so each becomes worth ten times as much. Multiplying by 100 does it twice, and by 1,000 three times. Dividing moves them the same way to the right. The decimal point does not move at all — the digits move past it.

For example, when 52.8 is multiplied by 100, the 5 moves from tens to thousands, the 2 moves from ones to hundreds, and the 8 moves from tenths to tens. The digits keep their order, but each one becomes worth one hundred times as much.

52.8 × 100 5,280

Interactive place-value machine Each digit in 52.8 moves two places left, producing 5,280. two places left

Follow the calculation

  1. The multiplier 100 contains two factors of 10, so each non-zero digit must move two places to the left.
  2. The 5 moves from the tens place to the thousands place; the 2 moves from the ones place to the hundreds place; and the 8 moves from the tenths place to the tens place.
  3. The ones place is now empty, so a zero holds it. Reading the completed number gives 5,280.

Every digit changes place at the same time. Following the place-value columns makes this movement visible and avoids unreliable shortcuts about adding zeros.

The place-value rule

Each step from one place-value column to the next represents a factor of 10. A digit in the hundreds place is worth ten times as much as the same digit in the tens place. In the tenths place, it is worth one tenth of its value in the ones place.

This pattern continues seamlessly across the decimal point, so whole numbers and decimals follow the same rule.

Multiplication moves digits left the magnitude becomes larger
Division moves digits right the magnitude becomes smaller
10

one place

100

two places

1,000

three places

The number of zeros in 10, 100 or 1,000 tells you how many factors of 10 are present—and therefore how many places the digits must move.

Multiplying by 10, 100 and 1,000

Begin by counting the zeros in the multiplier. This gives the number of places that every digit must move to the left.

It is also useful to predict the size of the result. Multiplication by 100 should make the magnitude one hundred times as large. If an answer remains close to the starting value, the digits have probably not moved far enough.

Example 1: calculate 6.47 × 100

  1. 6.47Start with 6 ones, 4 tenths and 7 hundredths.
  2. Move every digit two places left.Ones become hundreds, tenths become tens and hundredths become ones.
  3. 6.47 × 100 = 647The value is one hundred times as large.

Example 2: calculate 0.39 × 1,000

  1. 3 tenths → 3 hundredsThe 3 moves from the tenths place to the hundreds place.
  2. 9 hundredths → 9 tensAt the same time, the 9 moves three places from hundredths to tens.
  3. 0.39 × 1,000 = 390A zero holds the empty ones place.

Dividing by 10, 100 and 1,000

Division reverses the direction of movement. Count the zeros in the divisor, then move every digit that many places to the right.

A whole number may therefore produce a decimal result. This is not a special case: the ones, tenths and hundredths places all belong to one continuous place-value system.

Example 3: calculate 725 ÷ 10

  1. 7 hundreds → 7 tensThe new place is worth one tenth of the original place.
  2. 2 tens → 2 onesThe decimal point remains between the ones and tenths places.
  3. 725 ÷ 10 = 72.5The result has one tenth of the original magnitude.

Example 4: calculate 4 ÷ 100

  1. 4 ones → 4 hundredthsTwo factors of 10 make the place value one hundred times smaller.
  2. Write zeros in the empty places.Both the ones and tenths places must be held by zeros.
  3. 4 ÷ 100 = 0.04The placeholder zero prevents 4 being read as 4 tenths.

Using 0.1 and 0.01

The numbers 0.1 and 0.01 are powers of ten smaller than 1.

Multiplying finds a fraction of the starting value

Multiplying by 0.1 means finding one tenth of the starting number, which is equivalent to dividing by 10. Similarly, multiplying by 0.01 finds one hundredth, so it is equivalent to dividing by 100.

Calculate 6 × 0.1

  1. 0.1 means one tenth.
  2. One tenth of 6 is 6 ÷ 10.
  3. Therefore, 6 × 0.1 = 0.6.

Calculate 6 × 0.01

  1. 0.01 means one hundredth.
  2. One hundredth of 6 is 6 ÷ 100.
  3. Therefore, 6 × 0.01 = 0.06.

Division counts how many fractional pieces fit

The calculation 3 ÷ 0.1 asks a precise question: how many tenths are contained in 3?

3÷ 0.1= ?
1 whole 10 tenths
1 whole 10 tenths
1 whole 10 tenths
  1. Each whole contains 10 tenths.
  2. Three wholes contain 3 × 10 = 30 tenths.
  3. Therefore, 3 ÷ 0.1 = 30.

Dividing by 0.01 uses the same reasoning, but now the pieces are hundredths. Since one whole contains 100 hundredths, 3 ÷ 0.01 = 300.

These four relationships summarise the pattern:

× 0.1=÷ 10

find one tenth

× 0.01=÷ 100

find one hundredth

÷ 0.1=× 10

count the tenths

÷ 0.01=× 100

count the hundredths

Negative numbers

Negative numbers follow exactly the same place-value movement. Keep the negative sign, then move the digits in the usual direction:

−3.6 × 100 = −360

−48 ÷ 1,000 = −0.048

Common mistakes

  • Moving only the decimal pointThe decimal point is the fixed boundary between the ones and tenths places. Move the digits between place-value columns instead.
  • Always adding zerosZeros are not attached automatically: 3.4 × 10 = 34, not 3.40. Add a zero only when it must hold an empty place.
  • Forgetting placeholder zeros7 ÷ 100 = 0.07. Without the zero in the tenths place, the 7 would occupy the tenths place rather than the hundredths place.
  • Assuming division always makes a number smallerDividing by a positive number greater than 1 reduces the magnitude, but dividing by a positive number between 0 and 1 increases it. For example, dividing by 0.1 is equivalent to multiplying by 10.

Continue

Next lesson

Column addition

Adding integers and decimals in columns, lining up the point and carrying.