Floating Point Binary
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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
1
1
1
1
0
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: 000011
To calculate the new mantissa: shift the digits of the mantissa to the left until the first two characters are 01
Exponent: 1110 = -2
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
1
1
1
1
0
Not Normalised
Mantissa Calculation
Original Mantissa: 000011
Normalised Mantissa: 011000
The digits are shifted to the left by
3
offset =
3
Exponent Calculation
Original Exponent: 1110 = -2
Subtracting the offset from the exponent value: -5
Normalised Exponent: 1011
Denary Value
Value: 0.0234375
Mantissa
Exponent
0
●
1
1
0
0
0
1
0
1
1
Normalised
What is the Normalised Floating Point representation of the denary value
-0.041015625
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 = -0.000010101
Answer
Denary = -0.041015625
Fixed Point Binary Value = -0.00001
Mantissa = -010101
Mantissa = 101011
Exponent Value = -4
Exponent = 1100
Mantissa
Exponent
1
●
0
1
0
1
1
1
1
0
0
The following is the binary floating point representation of a number
Mantissa
Exponent
1
●
0
1
0
1
0
1
1
0
0
Calculate the denary equivalent of the number
Clue
Standard Form = -0.10110 x 2
-4
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
1
●
0
1
0
1
0
1
1
0
0
Answer
Exponent = 1100
Exponent Value = -4
Mantissa = 101010 (Two's Complement)
Mantissa = -010110 (Signed Value)
Standard Form = -0.10110 x 2
-4
Fixed Point Binary Value = -0.00001011
Denary Value =
-0.04296875
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
1
●
0
1
0
1
0
1
1
0
0
The following is an approximate binary floating point representation of the number 0.8
Mantissa
Exponent
0
●
1
1
0
1
0
0
0
0
0
Calculate the denary equivalent of the binary number
Clue
Standard Form = 0.11010 x 2
0
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
1
0
1
0
0
0
0
0
Answer: Denary Equivalent
Exponent = 0000
Exponent Value = 0
Mantissa = 011010 (Two's Complement)
Standard Form = 0.11010 x 2
0
Fixed Point Binary Value = 0.1101
Denary =
0.8125
Mantissa
Exponent
-1
1
2
1
4
1
8
1
16
1
32
-8
4
2
1
0
●
1
1
0
1
0
0
0
0
0
