Place value and powers of ten

Number

\(\displaystyle 352\)

What is the value of the digit five in this number?

Count places from the right: ones, tens, hundreds. The five is in the tens place

A digit and its value are different

A place tells you what one unit in that column is worth. The same digit can have a different value in a different place

Three hundreds

Three groups of one hundred have value three hundred

Three tens

Three groups of ten have value thirty

Three ones

Three groups of one have value three

Starting number

\(\displaystyle 327\)

The hundreds and ones digits stay fixed. Predict the change when the tens digit increases from two to three

Increase only the tens digit by one and check your prediction. Reset it to two, then increase only the hundreds digit by one. Compare the changes in the total. Finally set the tens digit to zero. The ones digit stays seven

What happens when the tens digit becomes zero?

Powers name repeated factors

In a power of ten, ten is the base. The small raised number is the exponent. For a positive whole-number exponent, it counts factors of ten, not a number to multiply ten by

One factor

The tens place has weight ten

Two factors

The hundreds place has weight one hundred

Three factors

The thousands place has weight one thousand

The ones place uses exponent zero

Move one place to the right: divide its weight by ten and reduce the exponent by one. This reaches a weight of one, not zero

Hundreds to tens

One hundred divided by ten is ten

Tens to ones

Ten divided by ten is one. We write the ones weight as ten to the power zero

Evaluate this power of ten

Follow the place weights from hundreds to tens to ones, dividing by ten each time

Build the number from its non-zero places

For four thousand three hundred and eight, keep the thousands, hundreds and ones contributions. The tens digit is zero; leaving out its term must not move the eight out of the ones place

Find each contribution

Four thousands, three hundreds and eight ones make the total

Use a power in every term

The exponent counts the places to the right of that digit. The ones digit has no places to its right

Write the whole sum

Use one non-zero digit times a power of ten for each occupied place. Omit zero terms. Any term order is allowed. You can type * for multiplication and ^ before the exponent

Short expansion check

The same pattern continues to larger places

Each step left multiplies the place weight by ten and increases the exponent by one. Keep all positions, even when some digits are zero

Beyond thousands

These are the ten-thousands, hundred-thousands and millions weights

Count positions, not non-zero digits

In eighty-three thousand and five, the three is in the thousands place and the five is in the ones place. The two zeros between them do not close the gap

Expansion check

Read each term, then add

Terms can appear in any order. Their powers, not their order on the page, determine the places

Find each value

The first term is four ones; the second is five ten-thousands

Add the contributions

The absent thousands, hundreds and tens places contribute zero

Reading check

Recover one missing contribution

An expansion has total eight thousand four hundred and five. The thousands and ones terms are shown; find the one omitted term

Evaluate what is shown

Eight thousands and five ones make eight thousand and five

Subtract from the complete total

The difference is the missing contribution

Name the place

Four hundred is four hundreds. Write its place weight as a power of ten

Missing check

Keep each digit in its place

A digit tells you how many; its power of ten tells you how much each is worth

Build an expansion

Pair each non-zero digit with its original place weight. Use exponent zero for ones

Read an expansion

Evaluate the contributions and add. Missing places contribute zero

Recover a term

Subtract the known contributions from the total, then express the difference as a place-value term