Number Systems (Cambridge (CIE) A Level Computer Science) : Revision Note

Number bases

What is a number base?

• A number base is the number of different digits or symbols a number system uses to represent values

• Each place in a number represents a power of the base, starting from the right

Denary

• Denary is a number system that is made up of 10 digits (0-9)

• Denary is referred to as a base-10 number system

• Each digit has a weight factor of 10 raised to a power, the rightmost digit is 1s (10⁰), the next digit to the left 10s (10¹) and so on

• Humans use the denary system for counting, measuring and performing maths calculations

• Using combinations of the 10 digits we can represent any number

• In this example, (3 x 1000) + (2 x 100) + (6 x 10) + (8 x 1) = 3268

• To represent a bigger number we add more digits

Binary

• Binary is a number system that is made up of two digits (1 and 0) 

• Binary is referred to as a base-2 number system

• Each digit has a weight factor of 2 raised to a power, the rightmost digit is 1s (2⁰), the next digit to the left 2s (2¹) and so on

• Each time a new digit is added, the column value is multiplied by 2

• Using combinations of the 2 digits we can represent any number

• In this example, Binary 1100 = (1 x 8) + (1 x 4) = 12

• To represent bigger numbers we add more binary digits (bits)

Hexadecimal

• Hexadecimal is a number system that is made up of 16 digits, 10 numbers (0-9) and 6 letters (A-F)

• Hexadecimal is referred to as a base-16 number system

• Each digit has a weight factor of 16 raised to a power, the rightmost digit is 1s (16⁰), the next digit to the left 16s (16¹)

• A quick comparison table demonstrates a relationship between hexadecimal and a binary nibble 

• One hexadecimal digit can represent four bits of binary data

Binary to denary & denary to binary

• Binary numbers can be converted into denary and vice-versa

• For example the 8-bit binary number 01101001 can be converted into denary using the following method:

• Therefore the 8-bit binary number 01101001 is 105 as a denary value

• To convert the denary number 101 to binary, we firstly write out binary number system

• Then we start at the left and look for the highest number that is less than or equal to 101 and if so, place a 1 in that column

• Otherwise, place a 0 in the column

• 128 is bigger than 101 and therefore we place a 0 in that column

• 64 is smaller than 101 so we place a 1 in that column

  • 101 - 64 = 37

  • This now means we have 37 left to find

• 32 is smaller than 37 so we place a 1 in that column

  • 37 - 32 = 5

  • This now means we have 5 left to find

• 16 is bigger than 5 and therefore we place a 0 in that column

• 8 is bigger than 5 and therefore we place a 0 in that column

• 4 is smaller than 5 so we place a 1 in that column

  • 5 - 1 = 1

  • This now means we have 1 left to find

• 2 is bigger than 1 and therefore we place a 0 in that column

• 1 is equal to the number we have left so we place a 1 in that column

• 64 + 32 + 4 + 1 = 101

  • Therefore the denary number 101 in binary is 01100101

Hexadecimal to denary & denary to hexadecimal

• To convert the denary number 163 to hexadecimal, start by dividing the denary value by 16 and recording the whole times the number goes in and the remainder

• 163 ➗16 = 10 remainder 3

• In hexadecimal the whole number = digit 1 and the remainder = digit 2

• Digit 1 = 10 (A)

• Digit 2 = 3

• Denary 163 is A3 in hexadecimal

• To convert the hexadecimal number 79 to denary, start by multiplying the first hexadecimal digit by 16

• 7 ✖ 16 = 112

• Add digit 2 to the result

• 112 + 9 = 121

• Hexadecimal 79 is 121 in denary

Binary to hexadecimal & hexadecimal to binary

• To convert the binary number 10110111 to hexadecimal, first split the 8 bit number into 2 binary nibbles

• For each nibble, convert the binary to it’s denary value

• (1 x 8) + (1 x 2) + (1 x 1) = 11 (B)

• (1 x 4) + (1 x 2) + (1 x 1) = 7

• Join them together to make a 2 digit hexadecimal number

• Binary 10110111 is B7 in hexadecimal

• To convert the hexadecimal number 5F to binary, first split the digits apart and convert each to a binary nibble

• Join the 2 binary nibbles together to create an 8 bit binary number

• Hexadecimal 5F is 01011111 in binary

Binary Coded Decimal (BCD)

What is Binary Coded Decimal?

• Binary coded decimal is a number system that uses 4 bit codes to represent each denary digit (0-9)

• To represent the denary number 2500 in BCD format would be:

  • 2500 = 0010 0101 0000 0000

• BCD can be stored in a computer as either half a byte (4 bits) or as two 4 bit codes joined to from a byte (8 bits)

Four single bytes

Two bytes

One's & two's complement

What is one's complement?

• One's complement is a method of representing both positive and negative numbers

• To turn a positive binary number in to a negative the positive binary number is inverted (0 becomes 1 and 1 becomes 0)

• The issue with one's complement is that it can have two representations for 0 (positive and negative)

What is two's complement?

• Two's complement is another method of representing both positive and negative numbers

• To turn a positive binary number into a negative:

  • Positive binary number is inverted

  • '1' is added to the right most bit

• Using two's complement the leftmost bit is designated the most significant bit (MSB)

• To represent negative numbers this bit must equal 1, turning the column value into a negative

• Working with 8 bits, the 128 column becomes -128

• 8 bit two's complement can represent values between 0111 1111 (+127) and 1000 0000 (-128)

• An alternative way of thinking about this is:

  • Starting from the right, keep all the bits the same up to and including the first 1

  • Flip the rest of the digits

    • 0101 1000 becomes...

    • 1010 1000

Key reasons to use two's complement

• Consistency

  • You don’t need different rules for signed and unsigned numbers

  • Whether you're adding +37 and +58, or −37 and +58, or −37 and −58, the binary addition process is the same

  • Two’s complement works for any combination of positive and negative values

• Hardware simplicity

  • CPUs are built to do simple binary addition

  • Two’s complement means:

    • The same adder circuit can handle everything

    • No special logic is needed to detect signs or subtract manually

• It still works, up to a point

  • As long as the sum doesn't exceed the max value (+127 in 8-bit), you can add two positive numbers with two’s complement without any issues