Place Value and Expanded Notation – Standard 3 Maths

November 3, 2026November 3, 2026 Aven Brisail2 Comments

What you will learn

Say the value of each digit in a number up to 9 999.
Write a number in expanded notation, and change expanded notation back into a numeral.
Describe the same number in different ways, such as 624 tens and 5 ones.

Every digit in a number has a job. In 7 777 all four digits are 7, but each 7 is worth a different amount. In this lesson you will find out why. You will learn to break numbers into parts and put them back together.

Place and value

The place of a digit is where it sits in the number. The value of a digit is how much it is worth in that place.

Thousands	Hundreds	Tens	Ones
7	7	7	7
7 000	700	70	7

In 7 777 the first 7 is worth 7 000. The next is worth 700, then 70, then 7. Each place is worth ten times the place to its right.

Worked example

What is the value of the digit 4 in 2 456?

Show me how

Find the place of the 4. Count from the right: ones, tens, hundreds.

The 4 is in the hundreds place.

4 hundreds = 400.

Answer: the value of the 4 is 400.

Expanded notation

Expanded notation shows a number as the sum of the values of its digits.

4 352 = 4 000 + 300 + 50 + 2

When a digit is zero, it is worth nothing, so we can leave it out: 5 060 = 5 000 + 60.

To go back to a numeral, put each part in its place. The parts can be in any order: 7 + 2 000 + 400 = 2 407.

Worked example

Write 3 000 + 40 + 5 as a numeral.

Show me how

3 000 means 3 thousands.

There is no hundreds part, so the hundreds digit is 0.

40 means 4 tens and 5 means 5 ones.

Answer: 3 045.

Watch out! A common mistake

Do not just join the digits together. 3 000 + 40 + 5 is not 345. Check each place: thousands 3, hundreds 0, tens 4, ones 5. The answer is 3 045.

One number, many ways

You can describe a number in more than one way. Look at 6 245:

6 thousands, 2 hundreds, 4 tens and 5 ones
62 hundreds and 45 ones
624 tens and 5 ones
6 245 ones

They all mean the same number! 62 hundreds is 6 thousands and 2 hundreds put together. 624 tens is all the tens in 6 245.

Worked example

How many tens are in 3 580?

Show me how

Cover the ones digit: 358 | 0.

The digits left over tell you the number of tens.

Answer: 358 tens. 358 tens = 3 580.

Try it

TRY ITPlace-value builder

Press + and − to change each column. The builder shows the expanded notation of your number.

TRY ITDigit detective

Choose a game and tap the right answer.

Let’s practise

Work out each one in your copybook. Open Hint if you are stuck, then open Show answer to check your working.

1. What is the value of the digit 8 in 3 841?

Hint

Which place is the 8 in?

Show answer

The 8 is in the hundreds place, so its value is 800.

2. Write 9 263 in expanded notation.

Hint

Write the value of each digit with + between.

Show answer

9 000 + 200 + 60 + 3

3. Write 4 000 + 500 + 7 as a numeral.

Hint

There is no tens part.

Show answer

4 507

4. Write 1 090 in expanded notation.

Hint

Leave out the zeros.

Show answer

1 000 + 90

5. Fill in the gaps: 5 372 = ____ thousands, ____ hundreds, ____ tens and ____ ones.

Hint

Read the digits from left to right.

Show answer

5 thousands, 3 hundreds, 7 tens and 2 ones.

6. How many hundreds are in 4 600?

Hint

Cover the tens and ones digits.

Show answer

46 hundreds.

7. (a) 30 + 6 000 + 2 = ____
(b) 800 + 7 000 = ____
(c) 9 + 50 + 3 000 = ____

Hint

Put the thousands first, then hundreds, tens and ones.

Show answer

(a) 6 032
(b) 7 800
(c) 3 059

8. Kiran says the 5 in 5 108 is worth 5. Is he right? Explain.

Hint

Look at the place of the 5.

Show answer

No. The 5 is in the thousands place, so it is worth 5 000.

Quiz

Lesson quiz: 10 questions

Answer every question, then press Check my answers. Score 8 or more to earn the Place Value Pro badge.

Recap

The value of a digit depends on its place: thousands, hundreds, tens or ones.
Expanded notation adds the values of the digits: 4 352 = 4 000 + 300 + 50 + 2.
Zeros are left out of expanded notation, but they must be written in the numeral.
One number can be described in many ways: 6 245 is 62 hundreds and 45 ones, or 624 tens and 5 ones.

Words to know

place value: The value a digit has because of its place in a number.
Example: In 2 456 the place value of 4 is hundreds.
value: How much a digit is worth.
Example: The value of the 4 in 2 456 is 400.
expanded notation: A way of writing a number as the sum of the values of its digits.
Example: 5 060 in expanded notation is 5 000 + 60.
base ten blocks: Blocks that show ones, tens, hundreds and thousands.
Example: We used base ten blocks to build 1 245.

Try it at home

Make digit cards from paper (0 to 9). Pick four cards, make a number and write it in expanded notation.
Mini project: make a place value chart on a sheet of paper with four columns. Use beans or buttons as counters to show three different numbers, and ask an adult to read them.

Test tips

Tips for class tests

Label the places above the digits (Th, H, T, O) before you answer.
In expanded notation, check that each part has the right number of zeros.
When changing expanded notation to a numeral, put a zero in any empty place.

For parents and teachers

After this lesson, your child should be able to:

give the value of any digit in a four-digit number;
write numbers in expanded notation and back again;
describe a number in different ways, such as 62 hundreds and 45 ones.

Watch for children who join digits (writing 3 000 + 40 + 5 as 345). Ask them to say the value of each part aloud. A place value chart drawn on paper helps a lot.

Quiz scores and badges are saved only in this browser on this device. Nothing is sent anywhere. If the browser data is cleared, or another device is used, the badges will be lost.

Previous: Numbers to 10 000: Count, Read and Write – Standard 3 Maths
Next: Compare and Order Numbers – Standard 3 Maths
Course: Standard 3 Maths

Updates

2026/11/03
– First version of this lesson.

Attributions to media used in this post:

Dienes Material (Dyskalkulie).jpg – Rhetos – CC0 1.0 (via Wikimedia Commons)

ZR1000 (Dienes-Material).jpg – Rhetos – CC0 1.0 (via Wikimedia Commons)

Number Bases: Place Value, Expanded Form, Binary and Converting to Base 10

October 24, 2026October 25, 2026 Aven Brisail2 Comments

FORM 2 • TERM 1 • LESSON 06

We write numbers using ten digits, 0 to 9, and each place is worth ten times the place to its right. That is called base 10 (or the denary system). It is not the only way to write numbers. In this lesson you will meet other number bases, find the place value of digits in them, write numbers in expanded form, learn why computers use binary (base 2), and convert numbers into base 10.

Learning outcomes

By the end of this lesson you should be able to:

2.1.3.1 – Explain what a number base is and recognise numbers written in different bases.
2.1.3.2 – Find the place value of a digit in a number written in any base.
2.1.3.3 – Write numbers in other bases in expanded form.
2.1.3.4 – Explain why binary (base 2) is used in computers and other digital devices.
2.1.3.5 – Convert numbers written in other bases to base 10.

What is a number base?

The base of a number system is how many digits it uses, and also the number you multiply by to move one place to the left.

Base 10 uses the digits 0–9. Place values: 1, 10, 100, 1,000, …
Base 5 uses the digits 0–4. Place values: 1, 5, 25, 125, …
Base 2 (binary) uses only 0 and 1. Place values: 1, 2, 4, 8, 16, …

In base b you can only use the digits 0 up to b − 1. So 2431 could be a base 5 number, but 2451 could not, because base 5 has no digit 5. We show the base with a small number written below and to the right, called a subscript: 1011₂ is “one zero one one, base two”. A number with no subscript is in base 10.

Place value in other bases

Each place is worth a power of the base. The place on the far right is always worth 1, because any base to the power 0 is 1.

Base	5th place	4th place	3rd place	2nd place	1st place
10	10⁴ = 10,000	10³ = 1,000	10² = 100	10¹ = 10	10⁰ = 1
5	5⁴ = 625	5³ = 125	5² = 25	5¹ = 5	5⁰ = 1
8	8⁴ = 4,096	8³ = 512	8² = 64	8¹ = 8	8⁰ = 1
2	2⁴ = 16	2³ = 8	2² = 4	2¹ = 2	2⁰ = 1

The place value of a digit is the digit times the value of its place. In 2431₅, the 3 is in the 5¹ place, so it is worth 3 × 5 = 15.

Expanded form

Writing every digit times its place value, added together, is called expanded form. Adding up the parts gives the number in base 10:

2431₅ = 2 × 5³ + 4 × 5² + 3 × 5¹ + 1 × 5⁰ = 250 + 100 + 15 + 1 = 366.

This is also exactly how you convert to base 10. You met expanded form for base 10 in Form 1 in place value and number names; it works the same way in every base.

TRY ITConvert to base 10

Type a number written in a base from 2 to 9 and choose its base. The table shows the place value of every digit, and the expanded form adds them up to give the number in base 10.

Number:

Base:

Binary and computers

A computer is built from millions of tiny electronic switches. Each switch has only two states: off or on. That fits base 2 perfectly, because binary has only two digits: 0 for off and 1 for on. A single binary digit is called a bit, and 8 bits make a byte.

With 8 bits you can write every number from 0 to 11111111₂ = 255, which is 256 different values.
Letters are stored as numbers. In the common ASCII code, capital A is 65, stored as 1000001₂.
Pictures are grids of tiny dots (pixels), and the colour of each pixel is stored as binary numbers. Music, video and text messages are all stored the same way.
Two states are reliable: it is easy for a circuit to tell “on” from “off”, even when the signal is weak.

Worked examples

Example 1: Which bases are possible?

Question: Which of these numbers could be written in base 4:
(a) 3021
(b) 1402
(c) 333?

Base 4 uses only the digits 0, 1, 2 and 3.
(a) 3021 uses 3, 0, 2, 1 – allowed. (b) 1402 has a 4 – not allowed. (c) 333 – allowed.

Answer: (a) and (c).

Example 2: Place value of a digit

Question: What is the value of the digit 6 in 3605₈? Give your answer in base 10.

Places from the right: 8⁰ = 1, 8¹ = 8, 8² = 64, 8³ = 512.
The 6 is in the 8² place.
6 × 64 = 384.

Answer: 384.

Example 3: Binary to base 10

Question: Convert 1011₂ to base 10.

Expanded form: 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰.
= 8 + 0 + 2 + 1.

Answer: 11.

Example 4: A longer binary number

Question: Convert 110101₂ to base 10.

Place values from the left: 32, 16, 8, 4, 2, 1.
Add the places with a 1: 32 + 16 + 4 + 1.

Answer: 53.

Example 5: Base 3 to base 10

Question: Write 2102₃ in expanded form and convert it to base 10.

2 × 3³ + 1 × 3² + 0 × 3¹ + 2 × 3⁰.
= 54 + 9 + 0 + 2.

Answer: 65.

Example 6: Base 8 to base 10

Question: Convert 247₈ to base 10.

2 × 8² + 4 × 8¹ + 7 × 8⁰.
= 128 + 32 + 7.

Answer: 167.

Common mistakes

Using a digit that is too big. Base 6 uses only 0 to 5, so “362₆” is fine but “372₆” is not a real number.
Starting the place values at the wrong end. The place on the right is always worth 1.
Thinking b⁰ is 0. Any base to the power 0 is 1, so the last digit is just itself.
Reading 101₂ as “one hundred and one”. Say the digits: “one zero one, base two”. Its value is 5.
Forgetting the subscript. Without it, everyone will read the number as base 10.

Practice questions

Try these yourself first, then open each answer to check your working.

1. Which of these are not possible numbers in base 3? 1201, 2130, 22, 1003.

Show answer

2130 and 1003 (they use the digit 3, but base 3 only uses 0, 1 and 2).

2. Write down the place values of the first four places (from the right) in
(a) base 4
(b) base 7.

Show answer

(a) 1, 4, 16, 64
(b) 1, 7, 49, 343.

3. In base 10, what is the value of
(a) the digit 4 in 2413₅
(b) the 1 on the far left of 10110₂?

Show answer

(a) 4 × 25 = 100
(b) 1 × 16 = 16.

4. Write in expanded form:
(a) 3021₄
(b) 11001₂.

Show answer

(a) 3 × 4³ + 0 × 4² + 2 × 4¹ + 1 × 4⁰
(b) 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰.

5. Convert to base 10:
(a) 111₂
(b) 10010₂
(c) 11111111₂.

Show answer

(a) 7
(b) 18
(c) 255.

6. Convert to base 10:
(a) 43₅
(b) 1234₅
(c) 305₆.

Show answer

(a) 20 + 3 = 23
(b) 125 + 50 + 15 + 4 = 194
(c) 108 + 0 + 5 = 113.

7. Convert to base 10:
(a) 64₈
(b) 2020₃
(c) 876₉.

Show answer

(a) 48 + 4 = 52
(b) 54 + 0 + 6 + 0 = 60
(c) 648 + 63 + 6 = 717.

8. Give two reasons why computers store information in binary.

Show answer

Any two of: the electronic switches inside have only two states (off and on), which match the digits 0 and 1; two states are easy to tell apart reliably; every kind of data (numbers, letters, pictures, sound) can be coded as 0s and 1s.

Quick check

1. Which digits are used in base 2?

Show answer

0 and 1.

2. What is 100₂ in base 10?

Show answer

3. What is the value of the right-hand place in any base?

Show answer

Previous: Index Form, Standard Form and Scientific Notation (Positive Powers)

Next: Adding and Subtracting in Any Base (and Converting to a Common Base)

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 1, Strand 2.1 Number Operations and Number Theory, Topic 2.1.3 Number Bases – Learning Outcomes 2.1.3.1 (the concept of number bases), 2.1.3.2 (place value in other bases), 2.1.3.3 (expanded form), 2.1.3.4 (why binary is used in computers) and 2.1.3.5 (converting to base 10).

Attributions to media used in this post:

Binary clock Swiss railway station.jpg – BBCLCD – CC BY-SA 4.0 (via Wikimedia Commons)

Updates

2026/10/23

– First version of this lesson.

2026/10/25

– Put each lettered part of the questions and answers on its own line.

Decimals: Place Value, Number Names, Comparing and Ordering (Tenths, Hundredths and Thousandths)

October 4, 2026October 25, 2026 Aven BrisailLeave a comment

Whole numbers count whole things, but real life is full of parts of a whole. A price tag says $12.75, a sprinter runs 100 m in 10.08 seconds, and a bag of flour has a mass of 2.25 kg. These numbers are decimals: the dot, called the decimal point, separates the whole-number part from the part that is less than one. In this lesson you will extend the place-value chart from Place Value and Number Names to the right of the point, read and write decimals in words, find the value of each digit, and compare and order decimals. Decimals are another way of writing the fractions you met in What Are Fractions?, so keep that lesson in mind.

Tenths, hundredths and thousandths

In our base-ten system, each place is worth ten times the place to its right. Going the other way, each place is worth one tenth of the place to its left. That pattern carries on past the ones place:

Split one whole into 10 equal parts: each part is one tenth, 110 = 0.1.
Split one whole into 100 equal parts: each part is one hundredth, 1100 = 0.01.
Split one whole into 1000 equal parts: each part is one thousandth, 11000 = 0.001.

Money is a good model. One dollar is 100 cents, so 10 cents is a tenth of a dollar ($0.10) and 1 cent is a hundredth of a dollar ($0.01). Length works too: a millimetre is one thousandth of a metre, so 1 mm = 0.001 m.

Showing a decimal in three ways

Concretely – with objects. Let a flat stand for 1, a rod for 0.1 and a small cube for 0.01. One flat, two rods and three cubes show 1.23.
Pictorially – with a drawing. Shade 23 squares of a 10 × 10 grid to show 0.23, or mark the point on a number line: 2.47 lies between 2.4 and 2.5, a little closer to 2.5.
Symbolically – with numerals and fractions: 0.3 = 310, 0.07 = 7100 and 0.125 = 1251000.

The decimal place-value chart

Hundreds	Tens	Ones	•	Tenths	Hundredths	Thousandths
	3	6	•	4	5	2
5	0	7	•	3	9	6

The numbers 36.452 and 507.396 on a place-value chart. There is no “oneths” place: the places are symmetrical about the ones place, not about the point.

The place value of a digit is the name of its column; its value is what it is worth. In 36.452 the digit 5 is in the hundredths place, so its value is 5 hundredths = 0.05. Writing a number as the sum of its digit values is called expanded form:

36.452 = 30 + 6 + 0.4 + 0.05 + 0.002 = 30 + 6 + 410 + 5100 + 21000

Reading and writing decimals in words

There are two common ways to say a decimal. Both are correct, and you should be able to match either one to the number and to the quantity it stands for.

Number	As a fraction name	Read digit by digit
2.5	two and five tenths	two point five
0.36	thirty-six hundredths	zero point three six
4.205	four and two hundred and five thousandths	four point two zero five
12.043	twelve and forty-three thousandths	twelve point zero four three

For the fraction name, read the digits after the point as one whole number, then say the name of the last place: 0.36 ends in the hundredths place, so it is “thirty-six hundredths”. When you read digit by digit, say each digit separately: 2.25 is “two point two five”, not “two point twenty-five”.

TRY ITDecimal place-value explorer

Type a decimal. The chart shows each digit in its place, and the explorer writes the number in words and in expanded form.

Type a decimal (0 to 999.999):

Place-value chart
Hundreds	Tens	Ones	•	Tenths	Hundredths	Thousandths
–	–	–	–	–	–	–

Tip: each place is worth ten times the place to its right, and one tenth of the place to its left.

Comparing and ordering decimals

To compare decimals, line up the decimal points and compare the digits place by place from the left, just as you do with whole numbers. If the numbers have different numbers of decimal places, fill the gaps with zeros so they are the same length – zeros at the end of a decimal do not change its value (0.5 = 0.50 = 0.500).

Which is bigger, 0.5 or 0.45? Write them as 0.50 and 0.45. Both have 0 ones; in the tenths place 5 is more than 4, so 0.5 > 0.45. A longer decimal is not automatically a bigger one. On a number line, the bigger number is always further to the right.

Ascending order means smallest to largest; descending order means largest to smallest.

Worked examples

Example 1: The value of a digit

Question: What is the value of the 7 in 23.075?

After the point the places are tenths (0), hundredths (7) and thousandths (5).
The 7 is in the hundredths place.

Answer: Its value is 7 hundredths, 7100 or 0.07.

Example 2: Words to figures

Question: Write “twelve and forty-three thousandths” in figures.

The whole-number part is 12.
Forty-three thousandths must end in the thousandths place, the third place after the point, so it needs three digits: 043.

Answer: 12.043 (not 12.43, which is twelve and forty-three hundredths).

Example 3: Ascending order

Question: Write 0.6, 0.06, 0.65, 0.605 and 0.56 in ascending order.

Fill with zeros: 0.600, 0.060, 0.650, 0.605, 0.560.
Compare as if they were whole numbers of thousandths: 600, 60, 650, 605, 560.
Smallest to largest: 60, 560, 600, 605, 650.

Answer: 0.06, 0.56, 0.6, 0.605, 0.65.

Example 4: A sprint final

Question: At a school sports day in San Fernando, four runners recorded 10.21 s, 10.12 s, 10.2 s and 10.09 s for the 100 m. List the times from fastest to slowest.

Fastest means the smallest time, so this is ascending order.
Fill with zeros: 10.21, 10.12, 10.20, 10.09.
Compare the hundredths parts: 09, 12, 20, 21.

Answer: 10.09 s, 10.12 s, 10.2 s, 10.21 s.

Example 5: Expanded form to a number

Question: Write 5 + 0.3 + 0.008 as a single decimal.

5 ones, 3 tenths, no hundredths and 8 thousandths.
Put a 0 in the empty hundredths place.

Answer: 5.308.

Common mistakes

Thinking a longer decimal is bigger. 0.45 has more digits than 0.5, but 0.5 = 0.50 is bigger.
Leaving out place-holding zeros. Forty-three thousandths is 0.043, not 0.43.
Reading 2.25 as “two point twenty-five”. Read the digits one at a time: two point two five.
Inventing a “oneths” place. The first place after the point is tenths.

Practice questions

Try these yourself first, then open each answer to check your working.

1. Write the place value and the value of the 9 in 507.396.

Show answer

Hundredths place; value 0.09 (9100).

2. Write 3.08 in words in two ways.

Show answer

Three and eight hundredths; three point zero eight.

3. Write 4 + 0.07 + 0.009 as a decimal.

Show answer

4.079.

4. Write 60.514 in expanded form.

Show answer

60 + 0.5 + 0.01 + 0.004.

5. Write 1.8, 1.08, 1.88 and 1.808 in descending order.

Show answer

1.88, 1.808, 1.8, 1.08.

6. Rainfall at three stations was 12.5 mm, 12.05 mm and 12.45 mm. Which station had the most rain?

Show answer

The one with 12.5 mm (12.50 is bigger than 12.45 and 12.05).

7. Which is greater: 9 tenths or 0.09? How many times greater?

Show answer

9 tenths = 0.9, which is ten times 0.09.

8. How many tenths are there in 2.7?

Show answer

27 tenths (2 wholes = 20 tenths, plus 7 tenths).

Quick check

1. What is the name of the third place after the decimal point?

Show answer

Thousandths.

2. Is 0.7 bigger or smaller than 0.69?

Show answer

Bigger: 0.70 > 0.69.

3. Write “five hundredths” as a decimal.

Show answer

0.05.

Next: Rounding Decimals, and Converting Between Fractions and Decimals

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 3, Strand 1.1 Number Operations and Number Theory, Topic 1.1.4 Decimals – Learning Outcomes 1.1.4.1 (representing decimals to thousandths concretely, pictorially and symbolically), 1.1.4.2 (matching number names to decimals and quantities), 1.1.4.3 (the place value and value of digits in decimals) and 1.1.4.4 (comparing and ordering decimals).

Attributions to media used in this post:

Base Ten Block.png – BarbourLA – CC BY-SA 4.0 (via Wikimedia Commons)

Place and value

Expanded notation

One number, many ways

Try it

Let’s practise

Quiz

Place value and expanded notation: worksheet

Part A: Let’s practise

Part B: Quiz

Answers (for the adult)

Recap

Words to know

Try it at home

Test tips

Updates

Attributions to media used in this post:

Learning outcomes

What is a number base?

Place value in other bases

Expanded form

Binary and computers

Worked examples

Example 1: Which bases are possible?

Example 2: Place value of a digit

Example 3: Binary to base 10

Example 4: A longer binary number

Example 5: Base 3 to base 10

Example 6: Base 8 to base 10

Common mistakes

Practice questions

Quick check

Attributions to media used in this post:

Updates

Tenths, hundredths and thousandths

Showing a decimal in three ways

The decimal place-value chart

Reading and writing decimals in words

Comparing and ordering decimals

Worked examples

Example 1: The value of a digit

Example 2: Words to figures

Example 3: Ascending order

Example 4: A sprint final

Example 5: Expanded form to a number

Common mistakes

Practice questions

Quick check

Attributions to media used in this post: