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.

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?

1. Base 4 uses only the digits 0, 1, 2 and 3.
2. (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.

1. Places from the right: 8⁰ = 1, 8¹ = 8, 8² = 64, 8³ = 512.
2. The 6 is in the 8² place.
3. 6 × 64 = 384.

Answer: 384.

Example 3: Binary to base 10

Question: Convert 1011₂ to base 10.

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

Answer: 11.

Example 4: A longer binary number

Question: Convert 110101₂ to base 10.

1. Place values from the left: 32, 16, 8, 4, 2, 1.
2. 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.

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

Answer: 65.

Example 6: Base 8 to base 10

Question: Convert 247₈ to base 10.

1. 2 × 8² + 4 × 8¹ + 7 × 8⁰.
2. = 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.

Quick check

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.

© 2026 Vedesh Kungebeharry. All rights reserved.