Number Base System – SS1 Mathematics Lesson Note (First Term)

A number base system writes numbers using place values that are powers of one fixed number, called the base.

• Base ten: place values are powers of ten.
• Other bases: two, seven, eight and so on use powers of their own base.
• Operations: numbers can be expanded, converted between bases, and added, subtracted, multiplied or divided in base two.

Behavioural objectives

By the end of the lesson, students should be able to:

1. expand numbers in base ten and in other bases using powers of the base;
2. convert numbers from other bases to base ten;
3. convert numbers from base ten to other bases;
4. convert numbers from one base to another through base ten;
5. add, subtract, multiply and divide numbers in base two.

What is a number base system?

A number base system is a way of counting in groups of one fixed size, and that size is the base.

• Everyday counting: people count in twos, fives, twenties and tens, and a day is counted as 24 hours.
• Base ten: people generally count in tens, using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
• Place value: the value of each digit depends on its position in the number.
• Base notation: a small number written below and to the right of a numeral shows its base, as in 645₇.
• Digits allowed: in base n, the digits run from 0 to n – 1.

Which digits are used in common bases?

What is place value in base ten?

Place value in base ten is the value a digit has because of its position, which is a power of ten.

• Units: 10⁰ = 1.
• Tens: 10¹ = 10.
• Hundreds: 10² = 100.
• Thousands: 10³ = 1000.

Examples

• 395₁₀ is 3 hundreds, 9 tens and 5 units.
• 395₁₀ = 3 × 10² + 9 × 10¹ + 5 × 10⁰ = 300 + 90 + 5
• 4075 = 4 × 10³ + 0 × 10² + 7 × 10¹ + 5 × 10⁰

The numbers are built on powers of ten, so this is the base ten number system.

How do you expand numbers in other bases?

To expand a number in base n, write each digit multiplied by n raised to the power of its place, starting from power 0 on the right.

• Base 8: uses powers of 8 (1, 8, 64, 512 and so on).
• Base 7: uses powers of 7 (1, 7, 49, 343 and so on).
• Base 2: uses powers of 2 (1, 2, 4, 8, 16, 32 and so on).

Example

Expand 645₇, 26523₇ and 101101₂.

1. 645₇ = 6 × 7² + 4 × 7¹ + 5 × 7⁰ = 6 × 49 + 4 × 7 + 5 × 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

Classwork

Expand the following:

1. 735₈
2. 1010011₂

How do you convert a number from another base to base ten?

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

Base ten is also called the denary scale.

1. Write the power of the base under each digit, starting with power 0 on the right.
2. Multiply each digit by its power of the base.
3. Add all the products.

Worked examples

Convert the following to base 10: (a) 27₈ (b) 11011₂

1. 27₈ = 2 × 8¹ + 7 × 8⁰ = 2 × 8 + 7 × 1 = 16 + 7 = 23
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

Classwork

Convert the following to base ten:

1. 101011₂
2. 2120₃

How do you convert a number from base ten to another base?

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 number.
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.

Worked example

Convert 53_ten to base two.

• Reading the remainders upwards: 53_ten = 110101₂.
• Check: 32 + 16 + 0 + 4 + 0 + 1 = 53.

How do you convert a number from one base to another?

To convert between two bases other than base ten, first change the number to base ten, then change it to the new base.

Example 1

Convert 1534_six to base eight.

• Step 1: convert 1534_six to base ten.
• 1534_six = 1 × 6³ + 5 × 6² + 3 × 6¹ + 4 × 6⁰ = 216 + 180 + 18 + 4 = 418_ten
• Step 2: convert 418_ten to base eight.

• Reading the remainders upwards, 418_ten = 642_eight.
• Thus, 1534_six = 642_eight.

Example 2: finding unknown bases

Determine the number bases x and y in the following simultaneous equations:

• 32_x – 12_y = 9_ten ………. (1)
• 23_x – 21_y = 4_ten ………. (2)

1. Change equation (1) to base ten: (3 × x¹ + 2 × x⁰) – (1 × y¹ + 2 × y⁰) = 9.
2. Simplify: 3x + 2 – y – 2 = 9, so 3x – y = 9 ………. (1a)
3. Change equation (2) to base ten: (2x + 3) – (2y + 1) = 4.
4. Simplify: 2x – 2y = 2, so x – y = 1 ………. (2a)
5. Subtract (2a) from (1a): 2x = 8, so x = 4.
6. Substitute x = 4 in (2a): 4 – y = 1, so y = 3.

Thus, x = 4 and y = 3.

Classwork

1. If x represents a base number in the following equations, what is the value of x?
   1. 315_x – 223_x = 72_x
   2. 405_x + 43_eight = 184_ten
2. Convert each of the following to the base indicated:
   1. 10401.11_seven to base eight
   2. 4836_sixteen to base twelve

How do you add in base two?

Binary addition follows the same column method as base ten addition, but a column total of 2 or more is carried in base two.

Which identities must you remember for addition?

• 0 + 0 = 0
• 0 + 1 = 1 and 1 + 0 = 1
• 1 + 1 = 10
• 1 + 1 + 1 = 11
• 1 + 1 + 1 + 1 = 100

Worked examples

Simplify the following:

1. 1110 + 1001 = 10111
2. 1111 + 1101 + 101 = 100001

Steps for 1110 + 1001, from the right:

1. 0 + 1 = 1. Write 1.
2. 1 + 0 = 1. Write 1.
3. 1 + 0 = 1. Write 1.
4. 1 + 1 = 10. Write 0 and carry 1 to the next place.
5. The answer is 10111.
6. Check in base ten: 14 + 9 = 23, and 10111₂ = 23.

How to read column totals:

• 11: take 1, carry 1.
• 10: take 0, carry 1.
• 100: take 0, carry 10.

Classwork

Simplify the following:

1. 1001 + 101 + 1111
2. 10101 + 111

How do you subtract in base two?

Binary subtraction follows the same column method as base ten subtraction, but a borrowed 1 is worth 2.

Which identities must you remember for subtraction?

• 0 – 0 = 0
• 1 – 0 = 1
• 1 – 1 = 0
• 10 – 1 = 1
• 11 – 1 = 10
• 100 – 1 = 11

Worked examples

Simplify the following:

1. 1110 – 1001 = 101
2. 101010 – 111 = 100011

Steps for 1110 – 1001, from the right:

1. 0 – 1 cannot be done. Borrow from the next place to make 10, and 10 – 1 = 1.
2. The next digit is now 0. Then 0 – 0 = 0.
3. 1 – 0 = 1.
4. 1 – 1 = 0.
5. The answer is 0101, which is 101.
6. Check in base ten: 14 – 9 = 5, and 101₂ = 5.

How do you multiply and divide in base two?

Binary multiplication uses only 0 and 1 as multipliers, and binary division uses repeated subtraction.

• Multiplication facts: 0 × 0 = 0, 1 × 0 = 0 and 1 × 1 = 1.
• Long multiplication: multiply by each digit, shift the line left, then add the lines using binary addition.
• Division: use the principle of subtraction, as in long division.

Worked examples

Simplify: 1. 1110 × 110 2. 110 ÷ 10

Solution 1: 1110 × 110

• Multiply by 0: 0000
• Multiply by 1, shifted one place left: 11100
• Multiply by 1, shifted two places left: 111000
• Add the three lines: 1110 × 110 = 1010100
• Check in base ten: 14 × 6 = 84, and 1010100₂ = 84.

Solution 2: 110 ÷ 10

• 10 goes into 11 once: 11 – 10 = 1.
• Bring down the last 0 to make 10. 10 goes into 10 once: 10 – 10 = 0.
• The quotient is 11, so 110 ÷ 10 = 11.
• Check in base ten: 6 ÷ 2 = 3, and 11₂ = 3.

Evaluation

1. Explain how to convert a number from base two to base ten.
2. List the steps for changing a base ten number to another base.
3. Convert 1101₂ to base ten.
4. Add 101₂ and 11₂.
5. State the value of 1 + 1 + 1 in base two.

Assignment

Part A: Number base conversions

Choose the correct answer from the letters a to c for questions 1 to 5.

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 120 base 10 to base 3. (a) 11110₃ (b) 1210₃ (c) 12110₃
5. Convert 25 base 10 to base 2. (a) 11001₂ (b) 1001₂ (c) 1100₂
6. Convert 2364₇ to base 10.
7. Convert 105₁₀ to base 2.

Part B: Binary operations

Choose the correct answer from the letters a to c for questions 1 to 3.

1. Express in base two the square of 11₂. (a) 1001 (b) 1010 (c) 1011
2. Find the value of (101)² in base two. (a) 1010 (b) 1111 (c) 11001
3. Multiply 100001₂ by 11₂. (a) 1001 (b) 1100011 (c) 10111
4. Convert the following to a binary number:
   1. 10_ten
   2. (10_ten)²
5. Calculate 110₂ × (1011₂ + 1001₂ – 101₂).
6. Multiply 34₅ by 22₅.