Floating Point Binary
Maths and Computing
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Floating Point Binary
Definition
Current size definition
Normalised floating point binary representation with a mantissa and exponent both in two's complement binary
Mantissa:
6
Exponent:
4
Valid range for positive values:
Binary:
0.000000001 to 1111100
Denery:
0.001953125 to 124
Valid range for negative values:
Binary:
-0.000000001 to -10000000
Denery:
-0.001953125 to -128
Change the size of the mantissa and exponent
Select the size of the mantissa
3
4
5
6
7
8
10
12
Select the size of the exponent
3
4
5
6
The following is the floating point representation of a number
Mantissa
Exponent
1
●
1
1
1
0
1
1
0
1
1
The value is not normalised. What is the normalised form of this binary number?
Mantissa
0
1
0
1
0
1
0
1
0
1
0
1
Exponent
0
1
0
1
0
1
0
1
Clue
Mantissa: 111101
To calculate the new mantissa: shift the digits of the mantissa to the left until the first two characters are 10
Exponent: 1011 = -5
Offset = The number of places that the Mantissa is shifted to the left
To calculate the new exponent: subtract the offset from exponent value
Mantissa
Exponent
1
●
1
1
1
0
1
1
0
1
1
Not Normalised
Mantissa Calculation
Original Mantissa: 111101
Normalised Mantissa: 101000
The digits are shifted to the left by
3
offset =
3
Exponent Calculation
Original Exponent: 1011 = -5
Subtracting the offset from the exponent value: -8
Normalised Exponent: 1000
Denary Value
Value: -0.0029296875
Mantissa
Exponent
1
●
0
1
0
0
0
1
0
0
0
Normalised
What is the Normalised Floating Point representation of the denary value
-1.9375
Mantissa
0
1
0
1
0
1
0
1
0
1
0
1
Exponent
0
1
0
1
0
1
0
1
Clue
Fixed Point Binary Value = -1.1111
Answer
Denary = -1.9375
Fixed Point Binary Value = -1.1111
Mantissa = -011111
Mantissa = 100001
Exponent Value = 1
Exponent = 0001
Mantissa
Exponent
1
●
0
0
0
0
1
0
0
0
1
The following is the binary floating point representation of a number
Mantissa
Exponent
0
●
1
0
0
0
1
0
1
1
1
Calculate the denary equivalent of the number
Clue
Standard Form = 0.10001 x 2
7
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
0
0
0
1
0
1
1
1
Answer
Exponent = 0111
Exponent Value = 7
Mantissa = 010001 (Two's Complement)
Standard Form = 0.10001 x 2
7
Fixed Point Binary Value = 1000100
Denary Value =
68
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
0
0
0
1
0
1
1
1
The following is an approximate binary floating point representation of the number 0.86
Mantissa
Exponent
0
●
1
1
1
0
0
0
0
0
0
Calculate the denary equivalent of the binary number
Clue
Standard Form = 0.11100 x 2
0
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
1
1
0
0
0
0
0
0
Answer: Denary Equivalent
Exponent = 0000
Exponent Value = 0
Mantissa = 011100 (Two's Complement)
Standard Form = 0.11100 x 2
0
Fixed Point Binary Value = 0.111
Denary =
0.875
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
1
1
0
0
0
0
0
0
