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Session Objectives #9. Binary Arithmetic & BCD. Starter: How many binary digits would be required for this hexadecimal code and what is the 8 bit binary equivalent: #FF9B22. Converting Binary to Hexadecimal. Denary 45 in binary is 32 + 8 + 4 + 1 or 128 64 32 16 8 4 2 1

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Binary arithmetic bcd

Binary Arithmetic & BCD

Starter:

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

#FF9B22


Converting binary to hexadecimal

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

= 2D

TASK: 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


Binary coded decimal bcd

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 10 1 0 1

Therefore 75 in BCD becomes: 01110101

Now convert the following denary into BCD:

39

58

97


Binary arithmetic

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 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0, carry 1

For example; Now try:

75 = 0 1 0 0 1 0 1 1 a) 01101101

+ + 01110001

14 = 0 0 0 0 1 1 1 0

0 1 0 1 1 0 0 1 = 89 b) 01111000

Carry 1 1 1 + 00110011


Sign magnitude

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 0 101 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.


2s complement

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


Subtraction in binary

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 10 1 0 1 1

-14 = 1 1 1 1 0 0 1 0

0 0 1 1 1 1 0 1 = 61

Carry 1 1

Now attempt

97 + 23

b) 43 – 58

Some more examples:Click Here

Why binary arithmetic – Watch this video


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