Number systems: Class 9 Computer Science

Key points

A number system is a way of writing numbers using a fixed set of symbols called digits. The base (or radix) of a system is the number of different digits it uses. Each digit has a value that depends on its place, so these are called positional number systems.

The four number systems used in computing are:

• Decimal: base 10, digits 0 to 9. This is the system we use daily.
• Binary: base 2, digits 0 and 1. This is the language of the computer.
• Octal: base 8, digits 0 to 7.
• Hexadecimal: base 16, digits 0 to 9 and letters A to F, where A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15.

Computers use binary because electronic circuits have two clear states: on and off (or high and low voltage). These two states are easy to build, store and detect, and they are less likely to be misread than ten different levels. One binary digit is called a bit. Four bits make a nibble and eight bits make a byte.

In any positional system the value of a digit is the digit multiplied by the base raised to the power of its position. Positions are counted from the right, starting at 0. For example, in the decimal number 352, the value is 3 × 10² + 5 × 10¹ + 2 × 10⁰ = 300 + 50 + 2.

Binary to decimal: multiply each bit by 2 raised to its position and add the results. Example: 1011 in binary.

• 1 × 2³ = 8
• 0 × 2² = 0
• 1 × 2¹ = 2
• 1 × 2⁰ = 1
• Total = 8 + 0 + 2 + 1 = 11, so (1011)₂ = (11)₁₀.

Decimal to binary: divide the number by 2 again and again and write the remainders. Read the remainders from bottom to top. Example: 13.

• 13 ÷ 2 = 6 remainder 1
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1
• Reading upward gives 1101, so (13)₁₀ = (1101)₂.

The same division method works for other bases. To convert a decimal number to octal divide by 8, and to hexadecimal divide by 16. Write the remainders from bottom to top, using letters A to F for remainders 10 to 15.

Binary, octal and hexadecimal are linked by grouping bits:

• Binary to octal: group the bits in threes from the right. Each group becomes one octal digit.
• Binary to hexadecimal: group the bits in fours from the right. Each group becomes one hex digit.
• To go back, replace each octal digit by 3 bits or each hex digit by 4 bits.
• Example: 11010110 grouped as 1101 0110 gives D6 in hex.

Hexadecimal is popular because it is a short way to write long binary numbers. One hex digit stands for exactly four bits, so a byte is just two hex digits. It is used for colour codes in web pages, memory addresses and MAC addresses.

Binary addition follows four rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + 1 = 11 (write 1, carry 1)

Negative numbers can be shown using complements. The 1's complement is found by changing every 0 to 1 and every 1 to 0. The 2's complement is the 1's complement plus 1. Computers use 2's complement to store negative integers and to do subtraction by addition.

Computers also store letters and symbols as numbers. ASCII (American Standard Code for Information Interchange) uses 7 bits to represent 128 characters, for example capital A has the code 65 and a has 97. Extended ASCII uses 8 bits for 256 characters. Unicode uses more bits and can represent characters of many languages including Urdu.

Study tip: Practise conversions every day because they carry sure marks. Always show the division steps or the power of 2 steps, and check the answer by converting back. Memorise the powers of 2 up to 2⁸ (1, 2, 4, 8, 16, 32, 64, 128, 256) and the hex letters, which will make working fast.

Definitions and flashcards

Say the meaning first, then open the card to check.

Formula sheet

Print this page

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Numericals with worked steps

Common mistakes

• Reading the remainders from top to bottom in decimal to binary conversion. They must be read from bottom to top.
• Starting the position count at 1 instead of 0. The rightmost digit has power 0.
• Grouping bits from the left. Groups of 3 or 4 bits must start from the right side.
• Forgetting the carry in binary addition, especially 1 + 1 + 1.
• Writing hex letters wrongly, for example taking B as 12. B is 11, and C is 12.

Practise this chapter

Practice questions and timed test Printable worksheet