Understanding Our Number System
1. Digits and Place Value
Learning outcomes
- I can identify the value of a digit based on its position in a number.
- I can explain the base-ten place value system.
- I can read and write numbers using place value language.
- I can identify periods such as ones, thousands, and millions.
- I can use place value to represent and interpret large numbers.
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.
Worked Example 1
What is the value of the digit 7 in 57,321?
Solution
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.
Worked Example 2
Write the following number in words.
8,105,046
Solution
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.
Worked Example 3
Write 43,207 in expanded form.
Solution
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:
- Compare the number of digits.
- If they have the same number of digits, compare the leftmost digit.
- 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.
Worked Example 4
Which number is greater?
- 3,456,781
- 3,465,781
Solution
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.
Common Mistakes
❌ Thinking a digit always has the same value.
✔ A digit's value depends on its position.
❌ Reading numbers without recognising periods.
✔ 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.
❌ Forgetting zeros in expanded form.
✔ 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.