School of Business, Economics and
Informatics
Department of Management
Binary Numbers 2
Eva Szatmari
e.szatmari@bbk.ac.uk
Learning Objectives
•
•
•
•
•
Recap BIN
Fractions in BIN
Two’s complement notation
Excess notation
Floating point notation
Video Tutorials
http://bbk.libguides.com/computerscience
/mathstutorials
http://www.bbk.ac.uk/mathstutorials/
https://www.youtube.com/playlist?list=PL2F
y-5oxIlb6cG15wxnhRdAmdskil_3-7
Warning: floating point notation!
Recap BIN
• What are BIN nos?
• How do we convert BIN to DEC?
• How do we convert DEC to BIN?
Fractions in BIN
Example 1:
11.101
Example 2:
1010.1101
Fractions in BIN
Example 1:
5¼
Example 2:
7¾
Example 3:
2 1 10
Fractions in BIN
a) 101.0011
b) 1100.101
c) 7 7 16
d) 12 3 4
Fractions in BIN
a) 101.0011 = 5 𝟑 𝟏𝟔
b) 1100.101 = 12 𝟓 𝟖
c) 7 7 16 = 111.0111
d) 12 3 4 = 1100.11
Two’s Complement Notation
Negative numbers in BIN?
Two’s complement
Sign bit, distinguishing between negative
and positive numbers
What is 0 represented as in two’s
complement notation?
Two’s Complement Notation
First fix the number of digits, e.g.: 4 bits
Find the positive BIN equivalent
Copy all digits from right to left till a 1
had been copied
Complement (invert/flip) the rest of the
digits
Two’s Complement Notation
4 bits two’s complement
Example 1:
Example 2:
-2
-4
6 bits two’s complement
Example 3:
Example 4:
100100
011001
Two’s Complement Notation
5 bits two’s complement:
a) – 8
b) – 13
c) 10101
d) 11010
Two’s Complement Notation
5 bits two’s complement:
a) – 8 = 11000
b) – 13 = 10011
c) 10101 = - 11
d) 11010 = - 5
Excess Notation
Another way of representing negative
numbers in BIN
What is 0 represented as in excess
notation?
What positive numbers look like?
What negative numbers look like?
How many digits do you use?
Excess Notation
4 bits excess:
Example 1:
-7
Example 2:
-5
Example 3:
4
Example 4:
2
Excess Notation
4 bits excess:
a) - 6
b) - 1
c) 5
d) 7
Excess Notation
4 bits excess:
a) - 6 = 0001
b) - 1 = 0111
c) 5 = 1101
d) 7 = 1111
Floating Point Notation
s e1e2e3 m1m2m3m4
What
What
What
What
does this sequence mean?
is s?
are e1e2e3?
are m1m2m3m4?
Floating Point Notation
Normalization
Example 1:
0.0011001
Exponent:
Example 2:
101.10010
Exponent:
Mantissa:
Mantissa:
Floating Point Notation
Step1: identify the sign bit
(0 for +, 1 for – numbers)
Step 2: find the mantissa
(move the radix point to 0.1… format,
record the first 4 digits after the point)
Step 3: find the exponent in 3 bits excess
notation
Step 4: record the number:
s e1e2e3 m1m2m3m4
Floating Point Notation
Example 1:
−1 1 4
Example 2:
−2 3 8
Example 3:
33 4
Example 4:
25 8
Floating Point Notation
a) −1 7
b) −5 1
8
2
Floating Point Notation
a) −1 7
b) −5 1
= 11011111
2 = 11111011
8
Floating Point Notation
Example 1:
01011001
Example 2:
10101101
Example 3:
10101100
Example 4:
11011100
Floating Point Notation
a) 10111011
b) 10011100
Floating Point Notation
a) 10111011 = − 𝟏𝟏 𝟑𝟐
b) 10011100 = − 𝟑 𝟑𝟐
Any Questions?
Contact Details
Eva Szatmari
0207 631 6254
Or
e.szatmari@bbk.ac.uk
GOOD LUCK