Binary Numbers: Number Base Conversions – JSS3 Mathematics Lesson Note (Third Term)

A number base is the number of digits a counting system uses, and each place value is a power of that base.

Behavioural objectives

1. Expand numbers in base ten and in other bases using place values.
2. Convert numbers from other bases to base ten.
3. Convert base ten numbers to other bases by repeated division.

What is a number base?

A number base is the number a counting system is built on, and it is also the number of digits the system uses.

• Base ten (denary): the everyday system, using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Other ways of counting: people also count in twos, fives and twenties, and a day is counted as 24 hours.
• Place value: each digit’s value depends on its position, and each position is a power of the base.
• Largest digit: in any base, the digits run from 0 up to one less than the base.

Which digits does each base use?

How do place values work in base ten?

• 395: 3 Hundreds, 9 Tens and 5 Units, that is, 3 × 10² + 9 × 10¹ + 5 × 10⁰.
• Value of 395: 300 + 90 + 5.
• 4075: 4 Thousands, 0 Hundreds, 7 Tens and 5 Units, that is, 4 × 10³ + 0 × 10² + 7 × 10¹ + 5 × 10⁰.
• Why it is base ten: the place values are powers of ten.

What are the place values in other bases?

How do we expand numbers in other bases?

To expand a number, write each digit multiplied by the base raised to the power of its place.

1. Write the digits of the number.
2. Number the places from the right, starting at power 0.
3. Multiply each digit by the base raised to its power.
4. Add the products with plus signs.

Worked example

Expand 647₈, 26523₇ and 101101₂.

1. 647₈ = 6 × 8² + 4 × 8¹ + 7 × 8⁰ = 6 × 64 + 4 × 8 + 7 × 1
2. 26523₇ = 2 × 7⁴ + 6 × 7³ + 5 × 7² + 2 × 7¹ + 3 × 7⁰
3. 101101₂ = 1 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰

• Remember: a power of 0 always equals 1, so the last digit is multiplied by 1.
• Value of (a): 384 + 32 + 7 = 423 in base ten.
• Value of (c): 32 + 0 + 8 + 4 + 0 + 1 = 45 in base ten.

How do we convert to denary scale (base ten)?

To convert from another base to base ten, multiply each digit by the base raised to the power of its place, then add the results.

1. Expand the number using the base and place values.
2. Work out each power.
3. Multiply each digit by its power.
4. Add all the products.

Worked examples

Convert the following to base 10.

1. 17₈
2. 11011₂

Solutions

1. 17₈ = 1 × 8¹ + 7 × 8⁰ = 1 × 8 + 7 × 1 = 8 + 7 = 15
2. 11011₂ = 1 × 2⁴ + 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 1 × 16 + 1 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 16 + 8 + 0 + 2 + 1 = 27

How do we convert from base ten to other bases?

To convert from base ten to another base, divide repeatedly by the new base and read the remainders upwards.

1. Divide the base ten number by the new base.
2. Write down the quotient and the remainder.
3. Divide the quotient by the new base again.
4. Continue dividing until the quotient is zero.
5. Start at the last remainder and read upwards to get the answer.

• Check: every remainder must be smaller than the new base.
• Check: convert your answer back to base ten to confirm it.

Worked examples

1. Convert 68₁₀ to base 6.
2. Convert 129₁₀ to base 2.

Solutions

1. 68₁₀ to base 6

Reading the remainders upwards: 68₁₀ = 152₆

2. 129₁₀ to base 2

Reading the remainders upwards: 129₁₀ = 10000001₂

What mistakes should students avoid?

• Reading remainders downwards: always read from the last remainder up to the first.
• Stopping early: keep dividing until the quotient is 0.
• Wrong digits: a digit can never be equal to or larger than the base, so 8 cannot appear in base 8.
• Wrong powers: start the powers at 0 on the right, not at 1.
• Missing subscript: always write the base as a subscript, for example 152₆.

Evaluation

Expanding numbers

Expand the following.

1. 4343
2. 101111₂

Converting to base ten

Convert the following to base ten.

1. 10100₂
2. 2120₃

Converting from base ten

1. Convert 569₁₀ to base 8.
2. Convert 100₁₀ to base 2.

General evaluation

Convert the following to base seven.

1. 405₁₀
2. 876₁₀

Then evaluate the following.

3. 538 – 314 + 412

Assignment

Reading assignment

• New Gen Math Book 3, pg 15-17
• Essential Mathematics for J.S.S.3, pg 5-9

Weekend assignment

1. Express 342₆ as a number in base 10. (a) 134 (b) 341 (c) 143
2. Change the number 10010₂ to base 10. (a) 1001 (b) 40 (c) 18
3. Express 100₁₀ in base 2. (a) 100100 (b) 1100100 (c) 11001
4. Convert 120₁₀ to base 3. (a) 11110₃ (b) 1210₃ (c) 12110₃
5. Convert 25₁₀ to base 2. (a) 11001₂ (b) 1001₂ (c) 1100₂

Theory

1. Convert 1264₈ to base 10.
2. Convert 105₁₀ to base 2.