Higher Computing Science Representing Real Numbers

2. What are Real Numbers?
Real Numbers include:
 Whole Numbers (like 1,2,3,4, etc)
 Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )
 Irrational Numbers (like π, √3, etc )
 Real Numbers can also be positive, negative or zero

3. What are Real Numbers?
Real Numbers include:
 Whole Numbers (like 1,2,3,4, etc)
 Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )
 Irrational Numbers (like π, √3, etc )
 Real Numbers can also be positive, negative or zero
15 3148
-272/7
99/10014.75
0.000123
100.159 π
√3
340.1155
22.9
In Computing, real numbers are also known as floating point numbers

4. Binary Real Numbers
What do real numbers look like in binary?
In binary, the numbers after the decimal point have the following place
values:
1/2 1/4 1/8 1/16 1/32 1/64 1/128

5. Binary Real Numbers
What do real numbers look like in binary?
In binary, the numbers after the decimal point have the following place
values:
1/2 1/4 1/8 1/16 1/32 1/64 1/128
So, 8.5 could be represented as:
1000 before the decimal point (8)
and 1 after the decimal point (0.5)
8.5 = 1000.1

6. Binary Real Numbers
What do real numbers look like in binary?
In binary, the numbers after the decimal point have the following place
values:
1/2 1/4 1/8 1/16 1/32 1/64 1/128
So, 14.625 could be represented as:
1110 before the decimal point (14)
and 101 after the decimal point (0.625)
14.625 = 1110.101

7. Standard Form
Standard form is a scientific notation of representing numbers as a base
number and an exponent.

8. Standard Form
Standard form is a scientific notation of representing numbers as a base
number and an exponent.
Using this notation:
The decimal number 8674.26 can be represented as
8.67426 x 103
mantissa = 8.67426
base = 10
exponent = 3

9. Standard Form
Standard form is a scientific notation of representing numbers as a base
number and an exponent.
Using this notation:
The decimal number 753.34 can be represented as
7.5334 x 102
mantissa = 7.5334
base = 10
exponent = 2

10. Standard Form
Standard form is a scientific notation of representing numbers as a base
number and an exponent.
Using this notation:
The decimal number 0.000634 can be represented as
6.34 x 10-3
mantissa = 6.34
base = 10
exponent = -3

11. Standard Form
Standard form is a scientific notation of representing numbers as a base
number and an exponent.
Any number can be represented in any number base in the form m x be

12. Standard Form
Standard form is a scientific notation of representing numbers as a base
number and an exponent.
Any number can be represented in any number base in the form m x be
This also applies to binary numbers, which have a base of 2.
Binary numbers can be represented as m x 2e

13. Floating Point Notation
In floating point notation, the real number is stored as 2 separate bits of data
A storage location called the mantissa holds the complete number without
the point.

14. Floating Point Notation
In floating point notation, the real number is stored as 2 separate bits of data
A storage location called the mantissa holds the complete number without
the point.
A storage location called the exponent holds the number of places that the
point must be moved in the original number to place it at the left hand side.

15. Floating Point Notation
What is the exponent of 10110.110?
The exponent is 5, because the decimal point has to be moved 5 places
to get it to the left hand side.
The exponent would be represented as 0101 in binary

16. Floating Point Notation
How would 10110.110 be stored using 8 bits for the mantissa and 4 bits for
the exponent?
We have already calculated that the exponent is 5 or 0101.
10110.110 = 10110110 x 25
= 10110110 x 20101
It is not necessary to store the ‘x’ sign or the base because it is always 2.
Mantissa Exponent

17. Floating Point Notation
How would 24.5 be stored using 8 bits for the mantissa and 4 bits for the
exponent?
Remember, in binary, the numbers after the decimal point have the following
place values:
1/2 1/4 1/8 1/16 1/32 1/64 1/128
24 has the binary value 11000
0.5 (or 1/2) has the binary value .1
24.5 = 0011000.1

18. Floating Point Notation
How would 0011000.1 be stored using 8 bits for the mantissa and 4 bits for
the exponent?

19. Floating Point Notation
How would 0011000.1 be stored using 8 bits for the mantissa and 4 bits for
the exponent?
The exponent is 7 because decimal point has to move 7 places to the left
0011000.1 = 00110001 x 27
= 00110001 x 20111
Mantissa Exponent

20. Accuracy
Store 110.0011001 in floating point representation, using 8 bits for the
mantissa and 4 bits for the exponent.
Mantissa Exponent
The mantissa only holds 8 bits and so cannot store the last two bits
These two bits cannot be stored in the system, and so they are forgotten.
The number stored in the system is 110.00110 which is less accurate that its
initial value.

21. Accuracy
If the size of the mantissa is increased then the accuracy of the
number held is increased.
Mantissa (10 bits) Exponent

22. Range
If increasing the size of the mantissa increases the accuracy of the number
held, what will be the effect of increasing the size of the
exponent?
Using two bits for the exponent, the exponent can have the value 0-3
Mantissa Exponent (2 bits)
This means the number stored can be in the range
.00000000 (0)
to
111.11111 (7.96875)

23. Range
Increasing the exponent to three bits, it can now store the values 0-7
Mantissa Exponent (3 bits)
This means the number stored can be in the range
.00000000 (0)
to
1111111.1 (127.5)
If the size of the exponent is increased then the range of the
numbers which can be stored is increased.

24. Floating Point Notation
Advantage:
We can store very large and very small numbers using
a small number of bits
Disadvantage:
The accuracy of the number being represented is lost
because numbers get rounded

25. Credits
Higher Computing – Data Representation – Representation of Real
Numbers
Produced by P. Greene and adapted by R. G. Simpson for the City of
Edinburgh Council 2004
Adapted by M. Cunningham 2010
All images licenced under Creative Commons 3.0
• Happy Pi Day by Mykl Roventine