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
0
●
0
0
0
1
0
0
1
0
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: 000010
To calculate the new mantissa: shift the digits of the mantissa to the left until the first two characters are 01
Exponent: 0101 = 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
0
●
0
0
0
1
0
0
1
0
1
Not Normalised
Mantissa Calculation
Original Mantissa: 000010
Normalised Mantissa: 010000
The digits are shifted to the left by
3
offset =
3
Exponent Calculation
Original Exponent: 0101 = 5
Subtracting the offset from the exponent value: 2
Normalised Exponent: 0010
Denary Value
Value: 2
Mantissa
Exponent
0
●
1
0
0
0
0
0
0
1
0
Normalised
What is the Normalised Floating Point representation of the denary value
3
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 = 11
Answer
Denary = 3
Fixed Point Binary Value = 11
Mantissa = 011000
Exponent Value = 2
Exponent = 0010
Mantissa
Exponent
0
●
1
1
0
0
0
0
0
1
0
The following is the binary floating point representation of a number
Mantissa
Exponent
0
●
1
1
1
1
1
1
0
1
0
Calculate the denary equivalent of the number
Clue
Standard Form = 0.11111 x 2
-6
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
1
1
1
1
1
0
1
0
Answer
Exponent = 1010
Exponent Value = -6
Mantissa = 011111 (Two's Complement)
Standard Form = 0.11111 x 2
-6
Fixed Point Binary Value = 0.00000011111
Denary Value =
0.01513671875
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
1
1
1
1
1
0
1
0
The following is an approximate binary floating point representation of the number 0.25
Mantissa
Exponent
0
●
1
0
0
0
0
1
1
1
1
Calculate the denary equivalent of the binary number
Clue
Standard Form = 0.10000 x 2
-1
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
0
0
0
0
1
1
1
1
Answer: Denary Equivalent
Exponent = 1111
Exponent Value = -1
Mantissa = 010000 (Two's Complement)
Standard Form = 0.10000 x 2
-1
Fixed Point Binary Value = 0.01
Denary =
0.25
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
0
0
0
0
1
1
1
1
