Numbering Systems

1
Numbering Systems Computing

• Understanding Binary, Octal and Hexadecimal
  Numbering Systems

2
What is a computer?

• Universal Information Manipulator
• Information data

3
Data

• 2 ways to manipulate an store
• Analog
• Digital

4
Analog

• Continuous
• Infinite precision
• Limited accuracy

5
Digital

• Incremental (series of numbers)
• Limited precision
• High accuracy

6
Analog vs digital

• Precision
• Accuracy

7
Computers understand voltage

• On / Off

1
1
0
8
Why Binary?

• Computers are made of a series of switches
• Each switch has two states ON or OFF
• Each state can be represented by a number
• ON is represented by 1
• Off is represented by 0

9
Understanding Placeholders

• Each numbering system has placeholders
• The possible values of each place holder depends
  on the maximum number of single-digit numbers
  available for that numbering system
• In the Base-10 world (aka Decimal), there are ten
  possible digits that each placeholder can take
  (0-9)

Think of placeholders like an odometer in a car
10
Understanding Placeholders
Base-10 (Decimal)
Ones
Tens
Placeholder Name
Hundreds
1004
103
16
Value
Exponential Expression
1024
1013
1006
11
Understanding Placeholders
Base-2 (Binary)
Ones
Twos
Placeholder Name
Fours
41
20
11
n10 Value
n10 Exponential Expression
221
210
201
12
Converting Base-2 to Base-10
ON
OFF
ON
ON/OFF
OFF
ON
n10 Exponent
20
21
22
23
24
0
0
2
1
16
Calculation





1910
13
Converting Base-10 to Base-2 21
Consecutive Division

• Step 1
• Q R
• 21 / 2 10 1
• 10 / 2 5 0
• / 2 2 1
• / 2 1 0
• 1 / 2 0 1

14
Converting Base-10 to Base-2
2110

• STEP TWO Concatenate remainders beginning from
  the bottom up.
• Q R 10101
• 21 / 2 10 1
• 10 / 2 5 0
• / 2 2 1
• / 2 1 0
• 1 / 2 0 1

15
Converting Base-10 to Base-2
2110
101012

RESULT
16
Introducing Octal

• Computer scientists are often looking for
  shortcuts to do things
• One of the ways in which we can represent binary
  numbers is to use their octal equivalents instead
• This is especially helpful when we have to do
  fairly complicated tasks using numbers

17
Introducing Octal

• The octal numbering system includes eight base
  digits (0-7)
• After 7, the next placeholder to the right begins
  with a 1

18
Octal Placeholders
Base-8 (Octal)
Ones
Placeholder Name
Eights
Sixty-Fours
642
84
11
n10 Value
n10 Exponential Expression
822
814
801
19
Converting Base-2 to Base-8
100011001010012
STEP ONE Take the binary number and from right
to left, group all placeholders in triplets. Add
leading zeros, if necessary
010
001
101
100
001
20
Converting Base-2 to Base-8
100011001010012
214518

STEP TWO Convert each triplet to its
single-digit octal equivalent. (HINT For each
triplet, the octal conversion is the same as
converting to a decimal number)
010
001
101
100
001
2
1
1
5
4
21
Converting Base-10 to Base-8
483210
STEP ONE Divide the Base-10 number by eight.
DO NOT DIVIDE PAST THE DECIMAL POINT INSTEAD
INCLUDE A REMAINDER
604
r0
4832 / 8

22
Converting Base-10 to Base-8
483210
STEP TWO Divide the quotient of the previous
expression by eight. Again, do not divide past
the remainder. Repeat the process until you have
a quotient of 0
604
r0
4832 / 8

75
r4
604 / 8

9
r3
75 / 8

1
r1
9 / 8

0
r1
1 / 8

23
Converting Base-10 to Base-8
483210

113408
STEP THREE Look at the column of remainders.
Reading from bottom to top, you will see the
octal equivalent
604
r0
4832 / 8

75
r4
604 / 8

9
r3
75 / 8

1
r1
9 / 8

0
r1
1 / 8

24
Hexadecimal Numbering

• Sometimes, it is necessary to use a numbering
  system that has more than ten base digits
• One such numbering system, hexadecimal, is useful
  on the Web
• Hexadecimal number, a Base-16 numbering system,
  is used in specifying web colors

25
Hexadecimal Numbering

• Because the Base-16 numbering system uses more
  than 10 base digits, we need to adopt new symbols
  for the Base-16 equivalents of the Base-10
  numbers 10, 11, 12, 14 and 15. We use letters to
  do this

26
Converting Base-10 to Base-16
21410
STEP ONE Divide the Base-10 number by sixteen.
DO NOT DIVIDE PAST THE DECIMAL POINT INSTEAD
INCLUDE A REMAINDER
13
r6
214 / 16

27
Converting Base-10 to Base-16
21410
STEP TWO Convert both the quotient and the
remainder to their hexadecimal equivalents
13
r6
214 / 16

D
6
Hex Equivalents
28
Converting Base-10 to Base-16
21410

D616
STEP THREE The quotient represents the first
digit of the hexadecimal equivalent and the
remainder represents the second digit
13
r6
214 / 16

D
6
Hex Equivalents
29
Questions?