Conversion of Number Bases – JSS2 Computer Studies Lesson Note (First Term)

Number base conversion is the process of changing a number from one base to another.

• From decimal: divide repeatedly by the new base and read the remainders upwards.
• To decimal: multiply each digit by its positional weight (a power of the base) and add the results.

Behavioural objectives

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

1. convert decimal numbers to binary, octal and hexadecimal;
2. convert binary, octal and hexadecimal numbers, including fractions, to decimal;
3. use positional weights to work out the decimal value of a number;
4. name the common data storage units and state that a byte is 8 bits.

What is a number base?

A number base is the number of different digits a number system uses before it starts a new place.

• Subscript: the small number after a value shows its base, for example 37₁₀ or 100101₂.
• Hexadecimal letters: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.

What are positional weights?

A positional weight is the value of a digit’s place, written as a power of the base.

• Whole numbers: the place just left of the point has power 0, then 1, 2, 3 moving left.
• Fractions: the places right of the point have powers -1, -2, -3 moving right.
• Any base to the power 0: equals 1.

How do you convert between decimal and binary?

Binary is base 2, so conversion uses division by 2 and powers of 2.

How do you convert decimal to binary?

Divide the decimal number by 2 until the quotient is zero, then read the remainders from the bottom upwards.

1. Divide the number by 2.
2. Write down the quotient and the remainder (0 or 1).
3. Divide the quotient by 2 again.
4. Stop when the quotient is 0.
5. Read the remainders from the last to the first.

Example 1: Convert 37₁₀ to binary.

Solution

37₁₀ = 100101₂

Example 2: Convert 93₁₀ to binary.

Solution

93₁₀ = 1011101₂

How do you convert a decimal fraction to binary?

Convert the whole part by division, then convert the fractional part by repeated multiplication by 2.

1. Convert the whole part by dividing by 2.
2. Multiply the fractional part by 2.
3. Write down the whole-number digit (0 or 1) of the answer.
4. Multiply the remaining fraction by 2 again, until it becomes 0.
5. Read the digits downwards after the point, then join both parts.

Example 3: Convert 25.625₁₀ into a binary number.

Solution

First convert the whole part, 25₁₀.

25₁₀ = 11001₂

Now convert the fractional part, 0.625.

0.625₁₀ = 0.101₂

Therefore, 25.625₁₀ = 11001.101₂

How do you convert binary to decimal?

Multiply each binary digit by its power of 2, then add the results.

1. Write the weight (power of 2) under each digit.
2. Multiply each digit by 2 raised to its weight.
3. Add all the products.

Example 1: Convert 10110₂ into a decimal number.

Solution: The binary number given is 1 0 1 1 0₂

Positional weights: 4 3 2 1 0

1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 0 × 2⁰

= 16 + 0 + 4 + 2 + 0 = 22₁₀

Example 2: Convert 11011₂ to a decimal number.

Solution: The binary number given is 1 1 0 1 1₂

Positional weights: 4 3 2 1 0

1 × 2⁴ + 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰

= 16 + 8 + 0 + 2 + 1 = 27₁₀

Example 3: Convert 1010.011₂ into a decimal number.

Solution: The binary number given is 1 0 1 0 . 0 1 1₂

Positional weights: 3 2 1 0 . -1 -2 -3

1 × 2³ + 0 × 2² + 1 × 2¹ + 0 × 2⁰ + 0 × 2^-1 + 1 × 2^-2 + 1 × 2^-3

= 8 + 0 + 2 + 0 + 0 + 0.25 + 0.125

1010.011₂ = 10.375₁₀

How do you convert between decimal and octal?

Octal is base 8, so conversion uses division by 8 and powers of 8.

How do you convert decimal to octal?

Repeatedly divide by 8, record each remainder (0 to 7) and read the answer upwards.

Example 1: Rewrite the decimal number 215₁₀ as an octal number.

Solution

215₁₀ = 327₈

Example 2: Convert decimal 1792₁₀ to an octal number.

Solution

1792₁₀ = 3400₈

How do you convert octal to decimal?

Write each digit place as a power of 8, multiply, then add the results.

Example 1: Convert 345₈ to a decimal number.

Solution: The octal number given is 3 4 5

Positional weights: 2 1 0

345₈ = 3 × 8² + 4 × 8¹ + 5 × 8⁰

= (3 × 64) + (4 × 8) + (5 × 1)

= 192 + 32 + 5 = 229₁₀

Example 2: Convert 3462₈ into a decimal number.

Solution: The octal number given is 3 4 6 2₈

Positional weights: 3 2 1 0

3 × 8³ + 4 × 8² + 6 × 8¹ + 2 × 8⁰

= 1536 + 256 + 48 + 2 = 1842₁₀

Example 3: Convert 362.35₈ into a decimal number.

Solution: The octal number given is 3 6 2 . 3 5

Positional weights: 2 1 0 . -1 -2

3 × 8² + 6 × 8¹ + 2 × 8⁰ + 3 × 8^-1 + 5 × 8^-2

= 192 + 48 + 2 + 0.375 + 0.078125

= 242.453125₁₀

How do you convert between decimal and hexadecimal?

Hexadecimal is base 16, so conversion uses division by 16 and powers of 16.

How do you convert decimal to hexadecimal?

Divide the decimal number by 16 repeatedly, change any remainder from 10 to 15 into A to F, and read the remainders upwards.

Example 1: Convert 1792₁₀ to hexadecimal.

Solution

1792₁₀ = 700₁₆

Example 2: Rewrite the decimal number 215₁₀ as a hexadecimal number.

Solution

215 ÷ 16 = 13, remainder 7

13 ÷ 16 = 0, remainder 13₁₀ = D

Therefore, 215₁₀ = D7₁₆

How do you convert hexadecimal to decimal?

Replace each letter with its value, multiply each digit by its power of 16, then add the results.

Example 1: Convert 42AD₁₆ into a decimal number.

Solution: The hexadecimal number given is 4 2 A D (A = 10, D = 13)

Positional weights: 3 2 1 0

4 × 16³ + 2 × 16² + 10 × 16¹ + 13 × 16⁰

= 16384 + 512 + 160 + 13

= 17069₁₀

Example 2: Convert 42A.12₁₆ into a decimal number.

Solution: The hexadecimal number given is 4 2 A . 1 2

Positional weights: 2 1 0 . -1 -2

4 × 16² + 2 × 16¹ + 10 × 16⁰ + 1 × 16^-1 + 2 × 16^-2

= 1024 + 32 + 10 + 0.0625 + 0.0078125

= 1066.0703125₁₀

How do computers store data?

Computers store information as 1s and 0s in storage such as memory, hard disks and USB drives.

• Bit: short for binary digit, the smallest unit of data a computer can store, either 1 (true) or 0 (false).
• Byte: the most common unit of digital data storage, equal to 8 bits.
• Kilobyte: 1 kilobyte is 1024 bytes, not 1000 bytes.

What are the data storage units?

The common data storage units, from smallest to largest, are the bit, byte, kilobyte, megabyte, gigabyte, terabyte and petabyte.

Evaluation

1. What is a bit? How many bits make a byte?
2. Convert 45₁₀ to binary.
3. Convert 10101₂ to a decimal number.
4. Convert 156₁₀ to an octal number.
5. Convert 3A₁₆ to a decimal number.

Assignment

Show full working for each conversion:

1. Convert 58₁₀ to binary.
2. Convert 1101.1₂ to a decimal number.
3. Convert 574₈ to a decimal number.
4. Convert 300₁₀ to a hexadecimal number.