A-Level Computing#BristolMet

Session Objectives#9

express numbers in binary, binary-coded decimal (BCD), octal and hexadecimal;

describe and use two’s complement and sign and magnitude to represent negative integers;

perform integer binary arithmetic, that is addition and subtraction

Create a GCSE calculator using an IF ELSE statement

A-Level Computing#BristolMet

Binary Arithmetic & BCD

Starter:

How many binary digits would be required for this hexadecimal code and what is the 8 bit binary equivalent:

#FF9B22

GCSE Computing#BristolMet

Converting Binary to Hexadecimal

Denary 45 in binary is 32 + 8 + 4 + 1 or

128 64 32 16 8 4 2 1 0 0 1 0 1 1 0 1

Split into 2 nibbles and treated as 4 bits each:

8 4 2 1 8 4 2 1 0 0 1 0 1 1 0 1

= 2 = 13

= 2DTASK: Using the same method convert a) 11101011 b) 10100011

Now try the reverse, convert to binary a) A5 b) 3B a) EB b) A3

a) 10100101 b)00111011

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Binary Coded Decimal (BCD)

Binary Coded Decimal (BCD) uses 4 bit binary to represent decimal digits 0 – 9.

You simply split the digits and treat the separately. For example, decimal 75 in BCD is as follows:

7 = 5 = 8 4 2 1 8 4 2 1 0 1 1 1 0 1 0 1

Therefore 75 in BCD becomes: 01110101Now convert the following denary into BCD:a) 39b) 58c) 97

A-Level Computing#BristolMet

Binary Arithmetic

Another reason why computers are designed to use binary is that addition is so simple in binary. In binary there are only 4 sums which need to be known:

0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 0, carry 1

For example; Now try:75 = 0 1 0 0 1 0 1 1 a) 01101101 + + 0111000114 = 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 = 89 b) 01111000Carry 1 1 1 + 00110011

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Sign & Magnitude

So far we have learned how to store positive whole numbers in binary but there is a need to use negative numbers and fractions.

You will remember that only 7 bits is need to represent the ASCII character set (127 characters) and this is the purpose of the 8th bit, to represent a +/-

For example: +/- 64 32 16 8 4 2 1 +75 = 0 1 0 0 1 0 1 1

-75 = 1 1 0 0 1 0 1 1

This is called sign/magnitude representation – the byte is in 2 parts, the sign (+/-) and the size of the number.

A-Level Computing#BristolMet

2s Complement

There are 2 problems with sign & magnitude representation. Firstly, the biggest number that can be represented is now halved – 127 instead of 255. Secondly, arithmetic now made more complicated as different bits means different things.

A solution to this is using a system called 2s complement – the last bit stands for -128. So the diagram looks like this:

-128 64 32 16 8 4 2 1

So -75 in 2s complement is:

-128 64 32 16 8 4 2 1 -75 = 1 0 1 1 0 1 0 1

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Subtraction in binary

Now using 2s complement subtraction is easier because 75 – 14 is the same as 75 + (-14)

-128 64 32 16 8 4 2 1 75 = 0 1 1 0 1 0 1 1 -14 = 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 1 = 61Carry 1 1

Now attempt a) 97 + 23b) 43 – 58

Some more examples:Click Here

Why binary arithmetic – Watch this video