SLR 10 & 11

Number Types

• N = Natural - Positive integers

• Z = Whole - Positive and Negative integers

• Q = Rational - values that can be represented as ratios, fractions or decimals - includes recurring numbers

• Irrational - cannot be expressed as a fraction - surds and endless decimals

• R = Real - rational and irrational numbers - can be used for measurement

• Ordinal - first, second, third

Bases

• Base 2 = binary

• Base 10 = denary

• Base 16 = hexadecimal

The base can be represented through a subscript value:

• 11₂ = 3 in binary

• 11₁₀ = 11 in denary

Why use hex?

• shorthand binary

• fewer mistakes

• easier to remember

• takes up less memory

Binary:

• Can be 16 bits

• Two types - signed and unsigned

• Signed can be negative (127 to -128)

• Use two's compliment on signed values

Two's Compliment

• Only use this if the binary number begins with a 1

• Flip the bit - 1 = 0 and 0 = 1

• Add 1 bit

Example Binary to Denary:

1011 0011

begins with a 1 so apply two's complement

1011 0011 - flip the bits

0100 1100 - add 1

0100 1101

convert to denary and make it negative

= 77

so 1011 0011 = -77

Example Denary to Binary:

- 99

99 in binary is 0110 0011

apply two's compliment

0110 0011 - flip the bits

1001 1100 - add 1 bit

1001 1101

so -99 = 1001 1101

Mathematical Operators:

Addition:

0 + 0 = 0

1 + 0 = 1

1 + 1 = (1) 0

0 + 1 = 1

1 + 1 + 1 = (1) 1

0100 1101 + 0011 1000 = 1000 0101

Subtraction:

To subtract, make the second value negative, then add

Ignore any overflow errors

0101 0110 - 0001 1111

make 0001 1111 negative:

1110 0000 + 1 = 1110 0001

0101 0110 + 1110 0001

Multiplication and Division:

In binary, to multiply/ divide you must double/ halve numbers.

To multiply, you will double and perform a left shift.

To divide, you will halve and perform a right shift.

Division example:

1001 0010 = 146

1001 0010 / 2

to halve you perform one right shift

0100 1001 =  73

1101 1000 = 216

1101 1000 / 4

to divide by 4 you perform two right shifts

0011 0110 = 54

Multiplication example:

0001 1101  * 4

to multiply by 4, perform two left shifts

0111 0100

0000 1111 * 6 = 0101 1010

to multiply by 6, multiply by 2 and by 4 then add

0000 1111 * 4 = 0011 1100

0000 1111 * 2 = 0001 1110

Fixed point and floating point

Fixed Point:

The column headings to the left of the point remain the same, the headings to the right halve

8  4  2  1  .  1/2  1/4  1/8  1/16

1100.1010 = 12.625

• not all decimals can be represented - rounding error

• limits the number of values, especially if signed

Floating Point:

Can store almost any decimal as the point can move

10110.011         0001

Sign - positive or negative

Mantissa - the main part of the number

Exponent - moves the floating point

A positive exponent moves to the right, a negative exponent moves to the left

Normalisation:

Standard form for binary - stores a number more efficiently - gets rid of excess 0 or 1

Ignore the exponent - will begin 0.1 (positive) or 1.0 (negative)

Example

1.111110110   1000

the point will move 5 places to the right

1.0110 - mantissa

exponent = exponent + -5 (5 places right) = 1000 + 1011

exponent = 10011

1.0110 10011

Denary to normalised binary

17.25 = 010001.01

Exp = 0000

TO NORMALISE:

010001.01 - move 5 to the left = -5

Exp = 5

Mantissa = 0.1000101

Exp = 0101