1.1 Number Systems

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Need to know 3 types:

1. Denary - base 10 number system

2. Binary - base 2 number system

3. Hexadecimal - base 16 number system

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Hexadecimals

Letter	Corresponding Denary Equiv	Corresponding Binary Equiv
A	10	1010
B	11	1011
C	12	1100
D	13	1101
E	14	1110
F	15	1111

To work out hex from binary, split up the binary number into sections of 4 bits each. To convert from denary to hex, work out the binary, then follow the above table.

Uses

• Error codes (memory locations of an error)

• ASCII

• Assembly Language

• URLs

• MAC addresses

• IPv6 addresses

• HTML colour codes

MAC

Stands for Media Access Control. It uniquely identifies a device on a network. It's made up of 48 bits, the first 3 hexadecimals are the manufacturers and the second half is the serial numbers.

HTML

Stands for HyperText Markup Language. It uses <tags> to bracket stuff. It is used to represent colours of text onscreen using RGB.

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Binary

Addition

If adding two digits:

Numbers	Answer	Carry Value
0 + 0	= 0	-
0 + 1	= 1	-
1 + 0	= 1	-
1 + 1	= 0	1

If adding three digits:

Numbers	Answer	Carry Value
0 + 0 + 0	= 0	-
0 + 0 + 1	= 1	-
0 + 1 + 1	= 0	1
1 + 1 + 1	= 1	1

Make sure to ADD the carry value to the sum. Otherwise, your answer will be incorrect.

Overflow

This occurs when bits are 'pushed' off the grid. An overflow error can occur.

As an example:

Adding two 8 bit numbers:

  0 1 1 0 1 1 1 0
  1 1 0 1 1 1 1 0
  ---------------
1 0 1 0 0 1 1 0 0

We have 9 bits in the answer.

Just using 8 bits in the above example would give us an incorrect answer. The generation of the 9th bit is a clear indication that the sum has exceeded the value of 255 as that is the maximum of an 8 bit number. This is an overflow error because an 8 bit computer cannot store that answer.

This is why calculators sometimes come up with an error when adding or multiplying stupidly large numbers.

Logical Shifts

Computers can do a logical binary shift to either the right or the left. When doing this, we can cause an overflow error if the last or first bit is a 1. The right most bit is referred to as the LSB (least significant bit) and left most bit is the MSB (most significant bit).It's important to note that doing a shift will CHANGE the value of the number.

If you do a shift to the left -> multiply the number by 2

If you do a shift to the right -> divide the number by 2

Two's Complement

This is used when we want negative numbers. We change the most left bit to a negative value. As an example, if we have an 8 bit number, we will change the MSB (128) to -128.

When applying this rule to a binary number, the left most bit always determines whether the number is positive or negative. If a 1 is there, the number is negative. If a 0 is there, the number is positive.

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Exam Questions

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