1. Floating Point Binary
A2 Computing
OCR Module 2509

2. Negative Binary Numbers
410 =
310 =
210 =
110 =
010 =
-110 = ?
Imagine a milometer… 9 9 9 9 9 9 9 9 1 4 3 2

3. Negative Binary Numbers
410 =
310 =
210 =
110 =
010 =
-110 =
-210 =
-310 =
-410 =
Imagine a milometer… 9 9 9 9 9 9 9 6 8 7 9
Two’s complement

4. Negative Binary Numbers: The Easy Method
Starting from the right, leave the digits alone up to the first 1
Change all the other digits from 1 to 0 or 0 to 1 = 1 1 1 1
Flip
Leave
sign
(0 ≡ positive; 1 ≡ negative) = 1 1
Two’s complement

5. Floating Point Binary 110100000 000011 0.1101 x 23 ≡ 110.1 ≡ 6.510
Imagine a 2 byte, 16 bit, number…
000011
mantissa
exponent
sign
The sign tells us it is a positive number (0 ≡ positive, 1 ≡ negative)
The mantissa and exponent tell us that that the number is x 23
x 23
≡ 110.1 ≡

6. Floating Point Binary: Negative Exponent
If the leading bit of the exponent is 1, the exponent is negative…
111110
mantissa
exponent
sign
N.B. If the exponent is negative, it will be written in two’s complement
Therefore, first convert the exponent from two complement to base 10:
= = -2
Number is therefore x 2-2
x 2-2 ≡
≡ 1/8+1/16+1/64 ≡

7. Normalisation of Floating Point Binary Numbers
In base 10:
234,567,000 ≈ x 1011
234,567,000 = x 109
This has lost some accuracy and precision
This number is normalised: it uses the mantissa to give the most accurate representation of the number by making sure the first digit after the decimal place is significant (i.e. not a zero)

8. Normalisation of Floating Point Binary Numbers
Normalise the floating point binary number
What do we know?
It is a positive number
It has a positive exponent
The number, as it stands, is x 22 =
Normalised: = x 2-1
We need the binary equivalent of -1 to represent the exponent
N.B. The exponent is negative so will need to be two’s complement
Two’s complement of -1 is
So, the normalised version of
is:
The sign bit is a 0
The leftmost bit of the exponent is also 0