Place Value

Introduction

Numbers are used every day to count, measure, compare, and solve problems. Whether reading the population of a country, the distance to another city, or the price of an item, understanding place value allows us to interpret numbers correctly.

Our number system is based on ten digits and follows a base-ten pattern. The position of each digit determines its value. Learning place value is one of the most important foundations in mathematics because it helps us read, write, compare, estimate, and calculate with numbers of any size.

Digits and Numbers

A digit is one of the ten symbols used to write numbers.

The ten digits are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Digits can be combined to form numbers.

For example:

• 7 is a one-digit number.
• 42 is a two-digit number.
• 583 is a three-digit number.
• 9,418 is a four-digit number.

Definition:
A digit is a single symbol (0–9) used to write numbers.

The Base-Ten Place Value System

Our number system is called the base-ten system because each place is worth ten times the place to its right.

For example:

• 10 ones = 1 ten
• 10 tens = 1 hundred
• 10 hundreds = 1 thousand
• 10 thousands = 1 ten thousand

This repeating pattern continues for larger numbers.

Notice how the value of each digit changes depending on its position in the number.

Place Value Chart

Place	Value
Millions	1,000,000
Hundred Thousands	100,000
Ten Thousands	10,000
Thousands	1,000
Hundreds	100
Tens	10
Ones	1

Each move to the left makes the value 10 times larger.

Each move to the right makes the value 10 times smaller.

The Value of a Digit

A digit's value depends on where it appears in the number.

Consider the number:

4,582

Digit	Place	Value
4	Thousands	4,000
5	Hundreds	500
8	Tens	80
2	Ones	2

Although the digit 5 is always the same symbol, its value changes depending on its position.

What is the value of the digit 7 in 57,321?

The 7 is in the thousands place.

Value:

7,000

Reading and Writing Numbers

Large numbers are separated into groups of three digits called periods.

For example:

5,472,318

can be divided into:

Period	Digits
Millions	5
Thousands	472
Ones	318

The number is read as:

Five million, four hundred and seventy-two thousand, three hundred and eighteen.

Understanding Periods

A period is a group of three digits.

Common periods include:

Period	Example
Ones	583
Thousands	24,583
Millions	7,024,583
Billions	4,007,024,583

Grouping digits into periods makes large numbers much easier to read.

 Write the following number in words: 8,105,046

Eight million, one hundred and five thousand, and forty-six.

Expanded Form

Expanded form shows the value of each digit separately.

Example:

6,483

Expanded form:

6,000 + 400 + 80 + 3

Expanded form helps us understand the contribution of each digit.

Write 43,207 in expanded form.

40,000 + 3,000 + 200 + 7

(The tens digit is 0, so it is not written.)

Representing Large Numbers

Place value helps us understand and compare large numbers.

Examples include:

• Population of Singapore: about 6 million people.
• Distance from Earth to the Moon: about 384,400 kilometres.
• A large stadium may hold 60,000 spectators.

Knowing place value helps us interpret the size and meaning of these numbers.

Comparing Numbers Using Place Value

To compare two numbers:

1. Compare the number of digits.
2. If they have the same number of digits, compare the leftmost digit.
3. Continue comparing each digit until they differ.

Example:

Compare:

482,715

479,999

Since:

482 thousand > 479 thousand,

482,715 is greater than 479,999.

Which number is greater?  3,456,781  OR  3,465,781?

Compare from left to right.

The millions are equal.

The hundred thousands are equal.

The ten thousands differ:

6 > 5

Therefore:

3,465,781 is greater.

Thinking a digit always has the same value.

Read numbers one period at a time.

Confusing the digit with its value.

Example:

In 8,531, the digit is 5, but its value is 500.

A zero means there is no value in that place.

Real-World Connection

Place value is used everywhere. Banks record balances that may reach millions of pounds or dollars, scientists work with very large and very small numbers, and governments use place value when reporting populations and national budgets. Understanding place value helps people read financial statements, maps, measurements, and statistical data accurately.

Did You Know?

The modern base-ten number system, including the use of zero as a place holder, developed from ideas originating in ancient India. It was later spread throughout the world by Arab mathematicians and eventually became the number system used almost everywhere today.

Key Terms

• Digit — one of the ten symbols (0–9) used to write numbers.
• Place value — the value of a digit determined by its position in a number.
• Base-ten system — a number system in which each place is worth ten times the place to its right.
• Expanded form — writing a number as the sum of the value of each digit.
• Period — a group of three digits in a large number.
• Ones, Tens, Hundreds — the first three place values in the base-ten system.

Key Takeaways

• Our number system uses ten digits and is based on the base-ten place value system.
• The value of a digit depends on its position within a number.
• Large numbers are organised into periods such as ones, thousands, and millions.
• Numbers can be written in standard form, word form, and expanded form.
• Understanding place value helps us read, write, compare, and interpret large numbers accurately.
• Place value is a fundamental mathematical concept used in everyday life, science, engineering, and finance.

Place Value

Introduction

Numbers can be written in different ways depending on the situation. You might see a number written using digits, written in words, or broken apart to show the value of each digit.

These different forms all represent the same number, but each one helps us understand numbers in a different way.

Learning to convert between these forms is an important skill because it strengthens your understanding of place value and helps you read, write, and compare large numbers.

Standard Form

Standard form is the way numbers are normally written using digits.

Examples:

• 7
• 45
• 382
• 6,194
• 425,018

Standard form is the quickest and most common way to write numbers.

Examples:

Standard Form
19
357
8,204
91,560

Word Form

Word form writes a number using words instead of digits.

Examples:

Standard Form	Word Form
15	fifteen
82	eighty-two
306	three hundred six
2,519	two thousand five hundred nineteen
45,830	forty-five thousand eight hundred thirty

Notice:

• Hyphens are used in numbers like twenty-one and sixty-eight.
• In most mathematics textbooks, the word "and" is not used in whole numbers. (It is often reserved for decimals.)

Example:

• 325 → three hundred twenty-five

rather than

• three hundred and twenty-five

Expanded Form

Expanded form shows the value of each digit according to its place value.

For example:

425

becomes

400 + 20 + 5

Each digit is written as its actual value.

Understanding Expanded Form

Consider the number:

4,582

Each digit has a different place value.

Digit	Place Value	Value
4	Thousands	4,000
5	Hundreds	500
8	Tens	80
2	Ones	2

Expanded form:

4,000 + 500 + 80 + 2

More Examples

Standard Form	Expanded Form
326	300 + 20 + 6
1,408	1,000 + 400 + 8
5,672	5,000 + 600 + 70 + 2
82,451	80,000 + 2,000 + 400 + 50 + 1

Notice that place values with a zero are usually omitted.

Example:

7,045

Expanded form:

7,000 + 40 + 5

The hundreds place is zero, so it is not included.

Converting Between Forms

Let's look at how to convert between the different forms.

Standard → Word

4,218

↓

Four thousand two hundred eighteen

Standard → Expanded

4,218

↓

4,000 + 200 + 10 + 8

Word → Standard

Seven thousand fifty-three

↓

7,053

Expanded → Standard

9,000 + 500 + 40 + 6

↓

9,546

Why Expanded Form is Useful

Expanded form helps us:

• understand place value
• compare numbers
• perform calculations
• estimate answers
• identify the value of each digit

For example:

6,482

Expanded form:

6,000 + 400 + 80 + 2

We can immediately see that:

• the digit 6 represents 6,000
• the digit 4 represents 400
• the digit 8 represents 80
• the digit 2 represents 2

Common Mistakes

Be careful not to confuse the value of a digit with the digit itself.

Example:

In the number 5,321

The digit 3 does not represent three.

It represents:

300

Another common mistake is forgetting place values containing zeros.

Example:

2,006

Expanded form:

2,000 + 6

Not:

2 + 6

Real-World Applications

Writing numbers in different forms is useful in many everyday situations.

Examples include:

• Writing cheques
• Reading prices
• Recording population figures
• Understanding bank statements
• Completing government forms
• Reading scientific data
• Interpreting statistics

Understanding expanded form also helps when learning addition, subtraction, multiplication, and division.

Worked Examples

Example 1

Write 3,504 in word form.

Answer:

Three thousand five hundred four

Example 2

Write 6,721 in expanded form.

Answer:

6,000 + 700 + 20 + 1

Example 3

Write:

Eight thousand ninety-six

in standard form.

Answer:

8,096

Example 4

Write:

9,000 + 300 + 50 + 7

in standard form.

Answer:

9,357

Example 5

What is the value of the digit 8 in 48,271?

Answer:

8,000

Did You Know?

The Hindu-Arabic number system that we use today was developed over many centuries and is based on place value. This system made calculations much easier than older numeral systems, such as Roman numerals, and became the standard numbering system used around the world.

Key Terms

Term	Definition
Standard Form	Writing a number using digits.
Word Form	Writing a number using words.
Expanded Form	Writing a number as the sum of the value of each digit.
Place Value	The value of a digit based on its position in a number.
Digit	Any of the symbols 0–9 used to write numbers.

Key Takeaways

• Standard form uses digits to write numbers.
• Word form uses words to describe numbers.
• Expanded form shows the value of each digit according to its place value.
• You should be able to convert easily between standard, word, and expanded forms.
• Expanded form helps explain how our base-ten place value system works.
• Understanding different number forms strengthens place value skills and prepares you for more advanced mathematics.

Dino Numbers

Introduction

Every day, we compare numbers without even thinking about it. We compare prices while shopping, scores in sports, distances on maps, and populations of cities.

To compare numbers correctly, we need to understand place value. The position of each digit tells us its value, allowing us to determine which number is larger or smaller.

Learning to compare and order numbers is an important skill that helps us solve problems in mathematics and everyday life.

Comparing Numbers

To compare two whole numbers, begin by looking at the digit with the greatest place value.

Ask yourself:

• Which number has more digits?
• If they have the same number of digits, compare the leftmost digits.
• Continue comparing each place value until you find a difference.

Step 1: Compare the Number of Digits

A number with more digits is always greater.

Examples:

Comparison	Larger Number
85 and 432	432
7,520 and 980	7,520
25,000 and 4,500	25,000

The first number has more place values, so it must be larger.

Step 2: Compare Place Values

If the numbers have the same number of digits, compare the digits from left to right.

Example:

4,382 and 4,197

Thousands:

4 = 4

Hundreds:

3 > 1

Therefore:

4,382 > 4,197

There is no need to compare the remaining digits because the hundreds place already determines which number is greater.

Another Example

Compare:

62,841 and 62,918

Ten-thousands:

6 = 6

Thousands:

2 = 2

Hundreds:

8 < 9

Therefore:

62,841 < 62,918

Understanding the Comparison Symbols

Three symbols are used when comparing numbers.

Symbol	Meaning	Example
>	Greater than	18 > 9
<	Less than	7 < 15
=	Equal to	42 = 42

A useful way to remember these symbols is:

The open side always points toward the larger number.

Tip: While some remember the symbols using the "hungry alligator" idea, understand that the open side always faces the larger value. This develops stronger mathematical reasoning.

Ordering Numbers

Ordering numbers means arranging them from smallest to largest or largest to smallest.

Least to Greatest

Arrange numbers starting with the smallest.

Example:

4,815

2,905

5,120

3,678

Correct order:

2,905

3,678

4,815

5,120

Greatest to Least

Arrange numbers starting with the largest.

Using the same numbers:

5,120

4,815

3,678

2,905

Using Place Value to Explain Your Answer

Good mathematicians explain why one number is greater than another.

Example:

Compare:

47,615

46,980

Explanation:

Both numbers have:

• 4 in the ten-thousands place.

Compare the thousands place:

• 7 > 6

Therefore:

47,615 is greater because it has more thousands.

Comparing Numbers with Zeros

Zeros still have place value.

Example:

Compare:

8,042

8,402

Thousands:

8 = 8

Hundreds:

0 < 4

Therefore:

8,042 < 8,402

Even though both numbers contain the same digits, their positions make the numbers different.

Real-World Applications

Comparing and ordering numbers is useful in many situations, including:

• Comparing prices while shopping.
• Ranking teams in sports.
• Comparing populations of cities.
• Ordering exam scores.
• Reading weather reports.
• Organizing scientific data.
• Comparing distances and travel times.

Worked Examples

Example 1

Compare:

5,430

5,403

Thousands:

5 = 5

Hundreds:

4 = 4

Tens:

3 > 0

Answer:

5,430 > 5,403

Example 2

Compare:

18,205

18,250

Ten-thousands:

1 = 1

Thousands:

8 = 8

Hundreds:

2 = 2

Tens:

0 < 5

Answer:

18,205 < 18,250

Example 3

Order from least to greatest.

7,415

6,980

7,401

6,875

Answer:

6,875

6,980

7,401

7,415

Example 4

Order from greatest to least.

12,405

12,540

12,450

12,504

Answer:

12,540

12,504

12,450

12,405

Example 5

Explain why:

52,801 > 52,198

Answer:

Both numbers have the same ten-thousands and thousands digits.

The hundreds digits are different:

8 > 1

Therefore, 52,801 is greater.

Did You Know?

Every year, countries conduct a population census to count the number of people living there. Governments compare and order these large numbers to help decide where to build schools, hospitals, roads, and other important services.

Key Terms

Term	Definition
Compare	To determine whether one number is greater than, less than, or equal to another.
Order	To arrange numbers according to their size.
Greater Than (>)	Indicates that one number is larger than another.
Less Than (<)	Indicates that one number is smaller than another.
Equal To (=)	Indicates that two numbers have the same value.
Place Value	The value of a digit based on its position within a number.

Key Takeaways

• Compare numbers by examining their place values from left to right.
• A number with more digits is always greater than a number with fewer digits.
• Use >, <, and = correctly when comparing numbers.
• Numbers can be arranged from least to greatest or greatest to least.
• Place value provides the mathematical reasoning behind every comparison.
• Understanding how to compare and order numbers is an essential skill used in mathematics and everyday life.

Rounding and Estimating

Introduction

Imagine you are shopping and want to know whether you have enough money to pay for several items. Instead of calculating the exact total, you can quickly estimate the cost by rounding each price.

Estimation is a valuable mathematical skill because it allows us to make quick decisions and check whether exact answers are reasonable.

One of the most common methods of estimating is rounding, where numbers are replaced with nearby values that are easier to work with.

What is Rounding?

Rounding means replacing a number with the nearest value at a specified place value.

For example:

• 47 rounded to the nearest ten is 50.
• 183 rounded to the nearest hundred is 200.

Rounding makes numbers easier to understand and calculate with.

The Rules for Rounding

Follow these simple steps.

Step 1

Identify the digit you are rounding to.

Step 2

Look at the digit immediately to its right.

• If it is 0, 1, 2, 3, or 4, keep the rounding digit the same.
• If it is 5, 6, 7, 8, or 9, increase the rounding digit by 1.

Step 3

Replace all digits to the right with zeros (for whole numbers).

Rounding Examples

Nearest Ten

Number	Rounded
23	20
47	50
81	80
95	100

Nearest Hundred

Number	Rounded
146	100
372	400
650	700
1,249	1,200

Nearest Thousand

Number	Rounded
2,450	2,000
2,750	3,000
8,320	8,000
14,680	15,000

Estimation

Estimation means finding an approximate answer instead of an exact one.

We often estimate by rounding numbers before performing calculations.

Example:

48 + 31

Round first:

48 → 50

31 → 30

Estimate:

50 + 30 = 80

Exact answer:

79

The estimate is very close.

Estimating Addition

Example:

398 + 205

Round:

398 → 400

205 → 200

Estimate:

400 + 200 = 600

Exact answer:

603

Estimating Subtraction

Example:

612 − 294

Round:

612 → 600

294 → 300

Estimate:

600 − 300 = 300

Exact answer:

318

Estimating Multiplication

Example:

48 × 19

Round:

48 → 50

19 → 20

Estimate:

50 × 20 = 1,000

Exact answer:

912

The estimate tells us the exact answer should be close to 1,000.

Estimating Division

Example:

398 ÷ 21

Round:

398 → 400

21 → 20

Estimate:

400 ÷ 20 = 20

Exact answer:

18.95

Again, the estimate is very close.

When is Estimation Useful?

Estimation is useful whenever an exact answer is not necessary.

Examples include:

• Estimating shopping costs.
• Planning a travel budget.
• Checking calculator answers.
• Estimating travel time.
• Predicting attendance at an event.
• Estimating construction materials.

Professionals often estimate before performing detailed calculations.

Is an Estimate Reasonable?

A reasonable estimate is close enough to the exact answer to be useful.

Suppose a calculator gives:

427 × 19 = 81

Is this reasonable?

Estimate first:

427 ≈ 400

19 ≈ 20

400 × 20 = 8,000

Since 81 is nowhere near 8,000, the calculator result is clearly incorrect.

Estimation is an excellent way to check for calculation errors.

Solving Real-World Problems

Example 1

A family buys groceries costing:

$18

$24

$13

Estimate the total.

Round:

20 + 20 + 10

Estimate:

$50

Exact total:

$55

Example 2

A school has:

487 students

Estimate to the nearest hundred.

Answer:

500 students

Example 3

A runner completes:

9.8 km

Estimate to the nearest kilometre.

Answer:

10 km

Common Mistakes

Be careful not to round the wrong digit.

Example:

Round 4,286 to the nearest hundred.

Correct:

4,300

Incorrect:

4,200

Always look at the digit immediately to the right of the place you are rounding.

Another common mistake is treating an estimate as an exact answer.

Remember:

An estimate is only an approximation.

Real-World Applications

Estimation and rounding are used every day in:

• Shopping
• Banking
• Engineering
• Construction
• Scientific investigations
• Travel planning
• Business
• Medicine

Professionals often estimate first to check whether detailed calculations are sensible.

Worked Examples

Example 1

Round 563 to the nearest hundred.

Answer:

600

Example 2

Round 7,462 to the nearest thousand.

Answer:

7,000

Example 3

Estimate:

91 + 208

Round:

90 + 200

Estimate:

290

Example 4

Estimate:

62 × 11

Round:

60 × 10

Estimate:

600

Example 5

Why is estimation useful?

Answer:

It allows us to make quick calculations and check whether exact answers are reasonable.

Did You Know?

Astronauts, engineers, and scientists often make quick estimates before using computers to perform detailed calculations. Estimation helps them recognize impossible answers and identify mistakes before they become serious problems.

Key Terms

Term	Definition
Estimate	An approximate answer or value that is close to the exact one.
Rounding	Replacing a number with the nearest value at a specified place value.
Nearest Ten	Rounding a number to the closest multiple of 10.
Nearest Hundred	Rounding a number to the closest multiple of 100.
Reasonable Answer	An answer that is close enough to the expected value to make sense.
Approximation	Another word for an estimate or a value close to the exact answer.

Key Takeaways

• Rounding replaces a number with a nearby value that is easier to use.
• To round correctly, look at the digit immediately to the right of the place value being rounded.
• Estimation uses rounded numbers to produce approximate answers.
• Estimates help solve everyday problems quickly and efficiently.
• Estimation is an important tool for checking whether exact calculations are reasonable.
• Rounding and estimation are widely used in mathematics, science, engineering, business, and everyday decision-making.

Multiply and Divide

Introduction

Our number system is based on the number 10. This is called the base-ten place value system.

Every time we move one place to the left in a number, the value becomes ten times larger. Every time we move one place to the right, the value becomes ten times smaller.

Recognizing these patterns makes it much easier to multiply, divide, estimate, and work with large and small numbers.

The Base-Ten Number System

Our number system uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The value of a digit depends on its position.

Each place is worth 10 times the place immediately to its right.

Place Value	Value
Thousands	1,000
Hundreds	100
Tens	10
Ones	1

Notice the pattern:

• 1 × 10 = 10
• 10 × 10 = 100
• 100 × 10 = 1,000
• 1,000 × 10 = 10,000

These are called powers of ten.

Powers of Ten

A power of ten is a number made by multiplying 10 by itself one or more times.

Power	Value
10⁰	1
10¹	10
10²	100
10³	1,000
10⁴	10,000
10⁵	100,000

Each power of ten is ten times larger than the previous one.

Multiplying by 10, 100, and 1000

When you multiply a whole number by a power of ten, every digit moves to the left.

• ×10 → move one place left.
• ×100 → move two places left.
• ×1000 → move three places left.

The number becomes larger because each digit moves into a place worth ten times as much.

Examples

Calculation	Answer
6 × 10	60
24 × 10	240
35 × 100	3,500
482 × 100	48,200
7 × 1000	7,000
125 × 1000	125,000

Notice that the digits move, not simply "zeros are added." Thinking about place value is more important than memorizing shortcuts.

Dividing by 10, 100, and 1000

When dividing by a power of ten, every digit moves to the right.

• ÷10 → move one place right.
• ÷100 → move two places right.
• ÷1000 → move three places right.

The number becomes smaller because each digit moves into a place worth one-tenth as much.

Examples

Calculation	Answer
80 ÷ 10	8
560 ÷ 10	56
4,300 ÷ 100	43
75,000 ÷ 1000	75
126,000 ÷ 1000	126

Understanding the Pattern

Look at the number 347.

Operation	Result
347	347
347 × 10	3,470
347 × 100	34,700
347 × 1000	347,000

Now divide:

Operation	Result
347	347
347 ÷ 10	34.7
347 ÷ 100	3.47
347 ÷ 1000	0.347

Each multiplication moves the digits one place to the left for every factor of 10.

Each division moves the digits one place to the right for every factor of 10.

Patterns in a Place Value Chart

Every place value is related to the next.

Place	Relationship
Ones → Tens	×10
Tens → Hundreds	×10
Hundreds → Thousands	×10
Thousands → Ten Thousands	×10

Moving the opposite direction means dividing by 10 each time.

Making Predictions Using Place Value

Understanding place value patterns helps us predict answers without lengthy calculations.

Example 1

If:

42 × 10 = 420

Then:

42 × 100 = 4,200

Example 2

If:

5,600 ÷ 100 = 56

Then:

5,600 ÷ 1000 = 5.6

Example 3

If a number is multiplied by 100 twice:

35 × 100 × 100

This is the same as:

35 × 10,000

Answer:

350,000

Recognizing these patterns makes calculations much faster.

Common Mistakes

One common mistake is thinking that multiplying by 10 simply means "adding a zero."

This works for many whole numbers, but not for all numbers.

For example:

6.5 × 10 = 65

The digits have moved one place to the left.

Similarly:

650 ÷ 10 = 65

The digits have moved one place to the right.

Thinking in terms of place value works for both whole numbers and decimals.

Real-World Applications

Understanding place value patterns is useful in many situations, including:

• Converting metric units.
• Estimating calculations.
• Working with money.
• Reading population data.
• Scientific measurements.
• Engineering calculations.
• Computer programming.
• Organizing large sets of data.

Worked Examples

Example 1

Calculate:

48 × 100

Answer:

4,800

Example 2

Calculate:

7,500 ÷ 100

Answer:

75

Example 3

Complete the pattern.

8

80

800

8,000

Answer:

Each number is multiplied by 10.

Example 4

What happens to the digits when a number is divided by 100?

Answer:

Each digit moves two places to the right, making the number one hundred times smaller.

Example 5

Predict:

250 × 1000

Answer:

250,000

Did You Know?

The metric system is based on powers of ten. This makes converting between units very simple. For example, 1 kilometre = 1,000 metres, 1 metre = 100 centimetres, and 1 kilogram = 1,000 grams. Because everything is based on powers of ten, metric conversions are much easier than converting between imperial units such as feet, inches, and miles.

Key Terms

Term	Definition
Place Value	The value of a digit determined by its position in a number.
Base-Ten System	A number system in which each place value is ten times the value of the place to its right.
Power of Ten	A number produced by multiplying 10 by itself one or more times.
Multiply	To increase a number by repeated groups or scaling.
Divide	To split a number into equal parts or reduce its size by a factor.

Key Takeaways

• Our number system is based on powers of ten.
• Every place value is ten times greater than the place to its right.
• Multiplying by 10, 100, or 1000 moves every digit to the left by one, two, or three places.
• Dividing by 10, 100, or 1000 moves every digit to the right by one, two, or three places.
• Understanding place value patterns makes calculations faster and helps predict answers.
• Place value patterns are used throughout mathematics, science, engineering, and the metric system.