Understanding Number Bases

🎯 By the end of this lesson...

You should be able to:

Explain what a number base is.
Explain why Base 10 is used in everyday counting.
Recognise number bases used in practical situations.
State which digits are allowed in a given base.
Determine whether a number is valid in a particular base.
Identify the place value of digits in different bases.
Convert numbers from another base to Base 10.
Convert whole numbers from Base 10 to another base.
Check whether a converted answer is reasonable.

🤔 Think About This...

💻

How does a computer count?

Humans normally count using ten digits:

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

A computer works using tiny electronic switches that can have only two states:

0 OFF

1 ON

Because only two states are available, computers use a number system containing only the digits 0 and 1.

This number system is called Base 2, or binary.

1️⃣ What Is a Number Base?

A number base tells us how many different digits are available in a number system before we move into a new place value.

In Base 10, we use ten digits:

0 1 2 3 4 5 6 7 8 9

After 9, no new single digit is available.

We therefore begin a new place value and write:

10

The symbols used in a number may look familiar, but their place values depend on the base being used.

2️⃣ Why Do We Normally Use Base 10?

The number system used in everyday life is called the decimal system, or Base 10.

It is widely believed that people began counting in groups of ten because humans have ten fingers.

✋ 5 fingers

+

✋ 5 fingers

=

🔟 10 fingers

Base 10 uses ten digits because it groups quantities in tens.

💡 Did You Know?

Different number bases appear in technology and everyday life.

💻

Base 2

Computers use binary digits: 0 and 1.

⏰

Time systems

Seconds and minutes are grouped in sixties.

🥚

Dozens

Eggs and other items may be grouped in twelves.

🧮

Base 8

Octal notation has been used in computing.

🎨

Base 16

Hexadecimal is commonly used for digital colour codes.

📦

Scores

One score means a group of twenty.

3️⃣ Writing the Base of a Number

A small number written at the lower right shows the base.

1011₂

A Base 2 number

243₅

A Base 5 number

572₈

A Base 8 number

3582₁₀

A Base 10 number

When no base is written, ordinary everyday numbers are normally assumed to be in Base 10.

4️⃣ Digits Allowed in Different Bases

The digits used in a base begin at 0 and end one number before the base.

Number base	Digits allowed	Highest digit
Base 2	0, 1	1
Base 3	0, 1, 2	2
Base 4	0, 1, 2, 3	3
Base 5	0, 1, 2, 3, 4	4
Base 8	0, 1, 2, 3, 4, 5, 6, 7	7
Base 10	0, 1, 2, 3, 4, 5, 6, 7, 8, 9	9

Highest digit = Base − 1

🕵️ Pattern Detective

Look at the highest digit in each base.

Base 2 1

Base 5 4

Base 8 7

Base 10 9

In every case, the highest available digit is one less than the base.

5️⃣ Is the Number Valid in Its Base?

Every digit in a number must be smaller than the base.

Example 1: Is 10101₂ valid?

Base 2 allows only 0 and 1.

1 0 1 0 1

All the digits are allowed, so 10101₂ is valid.

Example 2: Is 214₂ valid?

Base 2 allows only 0 and 1.

2 1 4

The digits 2 and 4 are not allowed in Base 2.

Therefore, 214₂ is invalid.

Example 3: Is 324₅ valid?

Base 5 allows 0, 1, 2, 3 and 4.

3 2 4

Every digit is less than 5, so 324₅ is valid.

🧠 StudyNest Method: Checking Whether a Number Is Valid

Step 1: Identify the base.

Step 2: Subtract 1 from the base to find the highest permitted digit.

Step 3: Check every digit in the number.

Step 4: If even one digit is equal to or greater than the base, the number is invalid.

6️⃣ Place Value in Base 10

Before learning place value in other bases, recall how Base 10 works.

Thousands	Hundreds	Tens	Ones
10³	10²	10¹	10⁰
1000	100	10	1
3	5	8	2

3582

= 3 × 1000 + 5 × 100 + 8 × 10 + 2 × 1

= 3000 + 500 + 80 + 2

7️⃣ Place Value in Other Bases

In any base, each place value is a power of that base.

Base 2

..., 2³, 2², 2¹, 2⁰

..., 8, 4, 2, 1

Base 5

..., 5³, 5², 5¹, 5⁰

..., 125, 25, 5, 1

Base 8

..., 8³, 8², 8¹, 8⁰

..., 512, 64, 8, 1

The rightmost place always has value 1 because any non-zero number raised to the power 0 equals 1.

8️⃣ Converting from Base 2 to Base 10

Multiply every digit by its Base 2 place value, then add the results.

Example 4: Convert 1011₂ to Base 10

Place value

8

4

2

1

Digit

1

0

1

Product

1 × 8

0 × 4

1 × 2

1 × 1

1011₂ = 8 + 0 + 2 + 1

= 11

1011₂ = 11₁₀

9️⃣ Converting from Another Base to Base 10

Example 5: Convert 243₅ to Base 10

The Base 5 place values are:

5², 5¹, 5⁰

25, 5, 1

Place value

25

5

1

Digit

2

4

3

Product

2 × 25

4 × 5

3 × 1

243₅ = 50 + 20 + 3

= 73

243₅ = 73₁₀

Example 6: Convert 572₈ to Base 10

572₈ = 5 × 8² + 7 × 8¹ + 2 × 8⁰

= 5 × 64 + 7 × 8 + 2 × 1

= 320 + 56 + 2

572₈ = 378₁₀

🧠 StudyNest Method: Converting to Base 10

Step 1: Identify the base.

Step 2: Write the place values as powers of the base, beginning with power 0 on the right.

Step 3: Multiply every digit by its place value.

Step 4: Add all the products.

Step 5: Write the answer in Base 10.

🔟 Converting from Base 10 to Another Base

To convert a Base 10 whole number to another base, repeatedly divide by the new base.

Example 7: Convert 13₁₀ to Base 2

13 ÷ 2 Quotient 6 Remainder 1

6 ÷ 2 Quotient 3 Remainder 0

3 ÷ 2 Quotient 1 Remainder 1

1 ÷ 2 Quotient 0 Remainder 1

Read the remainders from bottom to top:

1 1 0 1

13₁₀ = 1101₂

1️⃣1️⃣ Another Conversion from Base 10

Example 8: Convert 73₁₀ to Base 5

73 ÷ 5 Quotient 14 Remainder 3

14 ÷ 5 Quotient 2 Remainder 4

2 ÷ 5 Quotient 0 Remainder 2

Read the remainders from bottom to top:

2 4 3

73₁₀ = 243₅

The remainders become the digits of the new number. Always read them from the final remainder upwards.

🧠 StudyNest Method: Converting from Base 10

Step 1: Divide the Base 10 number by the required base.

Step 2: Record the quotient and remainder.

Step 3: Divide the quotient by the base again.

Step 4: Continue until the quotient becomes 0.

Step 5: Read the remainders from bottom to top.

Step 6: Write the required base beside the answer.

1️⃣2️⃣ Checking a Conversion

You can check an answer by converting it back to Base 10.

Example 9: Check that 13₁₀ = 1101₂

1101₂ = 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1

= 8 + 4 + 0 + 1

= 13

The conversion is correct.

🌍 Real-Life Number Bases

⏰ Time

Time does not use groups of ten throughout.

60 seconds make 1 minute.
60 minutes make 1 hour.
24 hours make 1 day.
7 days make 1 week.

These are examples of quantities grouped using different base-like systems.

🥚 Dozens

One dozen means 12 items.

3 dozen eggs = 3 × 12

3 dozen eggs = 36 eggs.

📦 Scores

One score means 20 items.

4 scores = 4 × 20

4 scores = 80 items.

🎯 Exam Tips

Check that every digit is smaller than the stated base.

When converting to Base 10, begin with power 0 on the right-hand side.

When converting from Base 10, record every remainder carefully.

Read repeated-division remainders from bottom to top.

Always write the base beside your final answer when it is not Base 10.

Check difficult conversions by converting your answer back to Base 10.

⚠ Common Mistakes

Mistake 1:

Using a digit that is not allowed in the stated base.

For example, the digit 2 cannot appear in Base 2.

Mistake 2:

Using powers of 10 when the number is written in another base.

Mistake 3:

Starting the rightmost place with power 1.

The rightmost place is always power 0.

Mistake 4:

Reading the remainders from top to bottom during repeated division.

Mistake 5:

Forgetting to write the base of the final answer.

Mistake 6:

Treating the base written below a number as an ordinary digit in the number.

🎮 Your Turn!

🟢 Foundation Questions

What does a number base tell us?
List all the digits allowed in Base 4.
List all the digits allowed in Base 6.
State the highest digit allowed in Base 9.
State whether 101101₂ is valid.
State whether 245₅ is valid.
State whether 736₈ is valid.
State whether 876₉ is valid.

🔵 Developing Questions

Convert 101₂ to Base 10.
Convert 1110₂ to Base 10.
Convert 132₄ to Base 10.
Convert 304₅ to Base 10.
Convert 247₈ to Base 10.
Convert 10₁₀ to Base 2.
Convert 18₁₀ to Base 3.
Convert 31₁₀ to Base 5.

🔴 Challenge Questions

Convert 110101₂ to Base 10.
Convert 4321₅ to Base 10.
Convert 456₇ to Base 10.
Convert 45₁₀ to Base 2.
Convert 100₁₀ to Base 4.
Convert 250₁₀ to Base 8.
A computer stores a value as 100101₂. What is its Base 10 value?
Determine the missing base:

23_b = 17₁₀

✅ Show Answers

It tells us how many digits are available before moving to a new place value.
0, 1, 2, 3
0, 1, 2, 3, 4, 5
8
Valid
Invalid because digit 5 is not allowed in Base 5.
Valid
Valid
101₂

= 1 × 4 + 0 × 2 + 1 × 1

= 5₁₀
1110₂

= 8 + 4 + 2 + 0

= 14₁₀
132₄

= 1 × 16 + 3 × 4 + 2

= 30₁₀
304₅

= 3 × 25 + 0 × 5 + 4

= 79₁₀
247₈

= 2 × 64 + 4 × 8 + 7

= 167₁₀
10₁₀ = 1010₂
18₁₀ = 200₃
31₁₀ = 111₅
110101₂

= 32 + 16 + 0 + 4 + 0 + 1

= 53₁₀
4321₅

= 4 × 125 + 3 × 25 + 2 × 5 + 1

= 586₁₀
456₇

= 4 × 49 + 5 × 7 + 6

= 237₁₀
45₁₀ = 101101₂
100₁₀ = 1210₄
250₁₀ = 372₈
100101₂

= 32 + 4 + 1

= 37₁₀
23_b means:

2b + 3 = 17

2b = 14

b = 7

🧠 Remember This

A base tells us how many digits are available in a number system.

The highest digit is always one less than the base.

Every digit in a valid number must be smaller than its base.

Place values are powers of the base.

To convert to Base 10, multiply digits by place values and add.

To convert from Base 10, repeatedly divide by the new base.

Read the remainders from bottom to top.

🎯 Before Moving On...

✅ Can I explain what a number base is?
✅ Can I identify digits allowed in a given base?
✅ Can I check whether a number is valid?
✅ Can I identify place values in different bases?
✅ Can I convert a number to Base 10?
✅ Can I convert a Base 10 number to another base?
✅ Can I check a conversion?
✅ Can I recognise bases in practical situations?

🎉 Well Done!

You can now understand number bases and convert numbers between Base 10 and other bases.

Remember:

Identify the base → Check the digits → Use the correct place values → Convert carefully → Check your answer

📈 Your Progress

Lesson 12 of 19

Real Numbers Progress 63%

Understanding Number Bases

🎬 See place values change with the base

🎯 By the end of this lesson...

🤔 Think About This...

How does a computer count?

1️⃣ What Is a Number Base?

2️⃣ Why Do We Normally Use Base 10?

💡 Did You Know?

Base 2

Time systems

Dozens

Base 8

Base 16

Scores

3️⃣ Writing the Base of a Number

4️⃣ Digits Allowed in Different Bases

🕵️ Pattern Detective

5️⃣ Is the Number Valid in Its Base?

Example 1: Is 10101₂ valid?

Example 2: Is 214₂ valid?

Example 3: Is 324₅ valid?

🧠 StudyNest Method: Checking Whether a Number Is Valid

6️⃣ Place Value in Base 10

7️⃣ Place Value in Other Bases

Base 2

Base 5

Base 8

8️⃣ Converting from Base 2 to Base 10

Example 4: Convert 1011₂ to Base 10

9️⃣ Converting from Another Base to Base 10

Example 5: Convert 243₅ to Base 10

Example 6: Convert 572₈ to Base 10

🧠 StudyNest Method: Converting to Base 10

🔟 Converting from Base 10 to Another Base

Example 7: Convert 13₁₀ to Base 2

1️⃣1️⃣ Another Conversion from Base 10

Example 8: Convert 73₁₀ to Base 5

🧠 StudyNest Method: Converting from Base 10

1️⃣2️⃣ Checking a Conversion

Example 9: Check that 13₁₀ = 1101₂

🌍 Real-Life Number Bases

⏰ Time

🥚 Dozens

📦 Scores

🎯 Exam Tips

⚠ Common Mistakes

🎮 Your Turn!

🟢 Foundation Questions

🔵 Developing Questions

🔴 Challenge Questions

🧠 Remember This

🎯 Before Moving On...

🎉 Well Done!

📈 Your Progress