Number System – SS1 Digital Technologies Lesson Note (First Term)

A number system is a way of counting and writing numbers using a fixed set of digits. Humans count in the decimal system (base 10), while computers use binary (base 2), because their switches are either on or off. Octal (base 8) and hexadecimal (base 16) are used to shorten long binary numbers.

Behavioural objectives

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

1. name the four main number systems and state the base of each;
2. explain why computers use binary;
3. state the decimal value of the hexadecimal letters A to F;
4. convert a decimal number to binary using the “divide by 2” method;
5. convert a binary number to decimal using place values;
6. group binary digits to write a number in octal or hexadecimal.

What are the key terms in number systems?

A number system is a method of representing numbers with a set of symbols called digits.

• Digit: a single symbol used to write a number, such as 7 or A.
• Base (radix): the number of different digits a system uses, including zero.
• Place value: the worth of a digit because of its position in the number.
• Bit: one binary digit, either 0 or 1 (short for “binary digit”).
• Byte: a group of 8 bits.

How do humans and computers count?

Humans count in decimal, while computers count in binary.

• Humans: we use the decimal system because we have ten fingers. We count from 0 to 9, and after 9 we combine digits to make 10.
• Computers: they have electronic switches that are either ON or OFF, so two digits are enough.

What are the four main number systems?

The four main number systems are decimal, binary, octal and hexadecimal.

1. Decimal (base 10): the system we use daily (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
2. Binary (base 2): the system computers use (0, 1).
3. Octal (base 8): used to shorten long binary strings (0, 1, 2, 3, 4, 5, 6, 7).
4. Hexadecimal (base 16): used for colours and memory addresses (0 to 9 and A to F).

What is binary?

Binary is a number system that uses only two digits: 0 and 1.

• Origin of the word: “binary” comes from “bi”, meaning two, as in a bicycle with two wheels.
• 0: means “Off” or “False”.
• 1: means “On” or “True”.
• Storage: every photo you take, every song you play and every text you send is stored as a long string of 1s and 0s.

What are the place values in binary?

Binary place values are powers of 2, starting from 1 on the right and doubling each step to the left.

What is hexadecimal?

Hexadecimal is a base 16 number system that uses the digits 0 to 9 and the letters A to F.

• Purpose: a long binary number such as 101101101 is hard to read, so hexadecimal gives a shorter form.
• Letters: after 9, the system uses letters for the values 10 to 15.
• Colours: web designers write colours in hex, for example FF0000 is red.

How do you convert decimal to binary?

To convert decimal to binary, divide the number by 2 repeatedly and record the remainders.

1. Divide the decimal number by 2 and write the remainder (0 or 1).
2. Divide the answer (quotient) by 2 again and write the remainder.
3. Repeat until the quotient is 0.
4. Read the remainders from the bottom to the top.

Example 1: convert 13 to binary

1. 13 ÷ 2 = 6 (remainder 1)
2. 6 ÷ 2 = 3 (remainder 0)
3. 3 ÷ 2 = 1 (remainder 1)
4. 1 ÷ 2 = 0 (remainder 1)

Reading the remainders from the bottom to the top gives 1101.

13₁₀ = 1101₂

Example 2: convert 22 to binary

1. 22 ÷ 2 = 11 (remainder 0)
2. 11 ÷ 2 = 5 (remainder 1)
3. 5 ÷ 2 = 2 (remainder 1)
4. 2 ÷ 2 = 1 (remainder 0)
5. 1 ÷ 2 = 0 (remainder 1)

The answer is 22₁₀ = 10110₂.

Note: Write the base as a small number at the bottom (a subscript) so that we know which system is used.

How do you convert binary to decimal?

To convert binary to decimal, multiply each digit by its place value and add the results.

1. Write the powers of 2 above the digits, from right to left.
2. Multiply each binary digit by its place value.
3. Add up the results.

Example: convert 1011 to decimal

(1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 8 + 0 + 2 + 1 = 11

So 1011₂ = 11₁₀.

How do you convert binary to octal and hexadecimal?

To convert binary to octal or hexadecimal, split the binary digits into groups and replace each group with one digit.

• Octal: group the bits in threes, starting from the right.
• Hexadecimal: group the bits in fours, starting from the right.
• Short groups: add zeros on the left to complete the first group.

Example 1: convert 101101 to octal

1. Group in threes: 101 and 101.
2. 101 = 5 and 101 = 5.
3. The answer is 55₈.

Example 2: convert 10101111 to hexadecimal

1. Group in fours: 1010 and 1111.
2. 1010 = 10, which is A. 1111 = 15, which is F.
3. The answer is AF₁₆.

How do decimal, binary and hexadecimal compare?

The table shows how some numbers look in each system.

Why does number system knowledge matter?

Number systems explain how a computer stores and shows data.

• Data storage: all data in a computer is held as binary digits.
• Programming: programmers meet hex codes when fixing computer errors or designing websites.
• Hardware: engineers use hex and octal to read memory addresses and shorten long binary numbers.

Evaluation

1. Why do computers use binary instead of decimal?
2. Convert the number 25 from decimal to binary.
3. What does the letter “C” represent in hexadecimal?
4. Convert 1110 from binary to decimal.
5. Write 110111 in octal by grouping the bits in threes.

Assignment

Answer all three parts in your notebook, showing your working.

1. Convert the decimal number 20 to binary.
2. Convert the binary number 1001 to decimal.
3. Write the decimal value of the hexadecimal letters B and E.