Place Value and Whole Numbers

What are whole numbers and place value?

Whole numbers are 0, 1, 2, 3, and so on: they do not include negative numbers or fractional parts. Place value tells you what a digit contributes to a number according to its position. In 72, the digit 7 contributes 70, not 7, because it occupies the tens place.

By the end of this lesson, you should be able to identify a digit's place and value, read whole numbers in words, convert between standard and expanded form, and explain which of two whole numbers is larger.

Counting numbers, zero, and digits

In this course, counting numbers means 1, 2, 3, and so on; whole numbers also include zero. Zero can describe a count of none: an empty box contains 0 pencils. A count of 12 pencils is a whole number, but a length of 12.5 centimetres is not a whole number.

A digit is one of the ten symbols 0 through 9. A number can be written with one digit or several digits. The number 606 uses three digit positions, even though the same symbol 6 appears twice. Its two sixes do not have the same value.

How the base-ten system works

Ten ones can be regrouped as one ten; ten tens as one hundred; and ten hundreds as one thousand. This grouping continues for larger numbers. Each place immediately to the left is worth ten times as much as the place immediately to its right.

The following chart shows the number 3,052,417. Read the places from right to left when identifying positions.

For any digit in a whole number:

\[\text{digit's contribution} = \text{digit}\times\text{place unit}.\]

For example, the 5 above contributes \(5\times10{,}000=50{,}000\). The digit is 5, its place is ten thousands, and its value in this number is 50,000. These are three different answers to three different questions.

Why zero matters

In 4,008, zero holds both the hundreds and tens positions. The number means four thousands and eight ones, not forty-eight. Although the zero contributions can be left out of an expanded sum, their positions cannot be deleted from standard form.

Compare 408, 4,008, and 4,080. In all three, the symbols 4 and 8 appear, but their positions change the numbers' values. Leading zeros are different: writing 0048 does not change the value 48. When counting significant positions in a whole number, ignore leading zeros.

Standard form, expanded form, and word form

Standard form is the usual digit notation, such as 61,309. Expanded form separates the contributions of the nonzero digits:

\[61{,}309=60{,}000+1{,}000+300+9.\]

You may also include the zero term \(0\times10\); omitting that term does not remove the tens position from the original number.

For word form, divide the digits into groups of three from the right. These groups are called periods: ones, thousands, millions, and so on. In 3 | 052 | 417, read each nonzero group and attach its period name: three million, fifty-two thousand, four hundred seventeen. The commas here separate thousands; they are not decimal points. This lesson uses English-language number notation.

Worked example: build a number from its parts

Problem: Write “seven hundred thousand, two thousand, and thirty-five” in standard form.

First translate the quantities: \(700{,}000+2{,}000+30+5\). Align their place positions before adding. There are no ten thousands or hundreds, so both positions need zeros in the final number.

\[700{,}000+2{,}000+30+5=702{,}035.\]

Check: Read 702,035 back as seven hundred two thousand, thirty-five. Its thousands group is 702 and its ones group is 035, matching the quantities in the question. Writing 72,035 would lose 630,000 because it puts the 7 in the wrong place.

Worked example: compare whole numbers

Problem: Which is larger, 58,207 or 58,270?

Both have five significant digit positions. Compare from the left: the ten-thousands digits match, the thousands digits match, and the hundreds digits match. The first difference is in the tens place: 0 tens versus 7 tens. Therefore,

\[58{,}207<58{,}270.\]

The ones digits cannot reverse this decision: the 70 contributed by seven tens exceeds any difference between two ones digits. As a numerical check, \(58{,}270-58{,}207=63\), a positive difference.

For nonnegative whole numbers without leading zeros, more digit positions means a larger number. If the position counts match, compare left to right until the first unequal pair. This rule is for whole numbers, not decimal digit strings.

Worked example: equal values in different groupings

Problem: A school stores 24 packets of 10 counters and 6 loose counters. How many counters are there?

There are \(24\times10=240\) counters in packets and 6 more loose counters, so the total is 246. Regrouping 24 tens gives 2 hundreds and 4 tens; the remaining 6 counters give 6 ones. Thus \(246=200+40+6\).

The grouping changes how the quantity is described, not how many counters exist. This is the same idea used when carrying in addition or regrouping in subtraction.

Check your understanding

Try these before reading the solutions. These are lesson self-checks; the separate Quizzes and Practice tabs are unchanged.

1. In 90,604, what is the place and value of the 6?
2. Write 305,018 in expanded form.
3. Write two million, forty thousand, nine in standard form.
4. Compare 109,980 and 110,008 using < or >. Identify the first different place.
5. A number has 4 thousands, 15 hundreds, and 2 ones. What is its standard form?

Solutions and reasoning

1. The 6 is in the hundreds place, so its value is \(6\times100=600\).
2. \(305{,}018=300{,}000+5{,}000+10+8\). The ten-thousands and hundreds contributions are zero.
3. \(2{,}000{,}000+40{,}000+9=2{,}040{,}009\). Keep all empty positions when writing standard form.
4. \(109{,}980<110{,}008\). The hundred-thousands digits match; the first difference is in the ten-thousands place, where 0 < 1.
5. \(4{,}000+1{,}500+2=5{,}502\). Fifteen hundreds regroup as one thousand and five hundreds. Do not concatenate 4, 15, and 2 as digit strings.

Common mistakes

• Naming the digit instead of its value: The digit 9 in 90,604 has value 90,000. “9” alone does not answer a question about its contribution.
• Deleting placeholder zeros: 305,018 is not 35,018. Keep empty positions in standard form.
• Starting comparisons at the right: Compare the largest places first. A larger ones digit does not necessarily make the entire number larger.
• Treating a comma as a decimal point: In this lesson, 2,040 means two thousand forty, not two and forty thousandths.
• Assuming every quantity is a whole number: Counts can be whole numbers; a measurement with a fractional part is not.
• Mixing up digit count and number size: Digit-count comparison works for whole numbers after removing leading zeros, not for arbitrary decimal representations.

Continue learning

• Rounding and Estimation: use place positions to select a nearby value.
• Addition and Subtraction: align equal place units and regroup when needed.
• Decimal Place Value: extend the same base-ten pattern to values smaller than one.

Sources and further reading

Public textbook topic outlines were consulted to check lesson scope. The explanations, numerical examples, table, and self-checks above were independently written for this lesson, not copied or closely paraphrased from textbook prose or exercises. No external artwork is reproduced.

• OpenStax Prealgebra 2e: Introduction to Whole Numbers — introductory whole-number coverage.
• OpenStax Prealgebra 2e: preface and reuse conditions — check current license restrictions before reusing textbook material.
• Siyavula: Working with Whole Numbers — another publicly accessible whole-number topic outline.
• Siyavula: open-textbook licensing — distinguishes adaptable unbranded CC BY editions from branded CC BY-ND editions.