CPS120: Introduction to
Computer Science
Computer Math: Converting to
Decimal
Binary
Decimal is base 10 and has 10
digits:
0,1,2,3,4,5,6,7,8,9
Binary is base 2 and has 2 digits:
0,1
For a number to exist in a given number system, the
number system must include those digits. For
example:
The number 284 only exists in base 9 and higher.
9
Codes
Given any positive integer base (RADIX)
N, there are N different individual symbols
that can be used to write numbers in the
system. The value of these symbols range
from 0 to N-1
All systems we use in computing are
positional systems

495 = 400 + 90 +5
Number Systems
We use the DECIMAL (10 system
Computers use BINARY (2 or some
shorthand for it like OCTAL (8 or
HEXADECIMAL (16
Power of 2 Number System
Binary
000
001
Octal
0
1
Decimal
0
1
010
011
100
101
2
3
4
5
2
3
4
5
110
111
6
7
6
7
100
10
8
1001
1010
11
12
9
10
16
Bases Higher than 10
How are digits in bases higher than 10
represented?
Base 16:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F
10
Conversions
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
Octal
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
Hex
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
Decimal Equivalents
Assuming the bits are unsigned, the decimal
value represented by the bits of a byte can be
calculated as follows:
1.
Number the bits beginning on the right using superscripts
beginning with 0 and increasing as you move left
•
2.
3.
4.
Note: 20, by definition is 1
Use each superscript as an exponent of a power of 2
Multiply the value of each bit by its corresponding power of
2
Add the products obtained
Converting Octal to Decimal
What is the decimal equivalent of the octal
number 642?
6 x 8² = 6 x 64 = 384
+ 4 x 8¹ = 4 x 8 = 32
+ 2 x 8º = 2 x 1 = 2
= 418 in base 10
11
Converting Hexadecimal to
Decimal
What is the decimal equivalent of the
hexadecimal number DEF?
D x 16² = 13 x 256 = 3328
+ E x 16¹ = 14 x 16 = 224
+ F x 16º = 15 x 1 = 15
= 3567 in base 10
Remember, base 16 is
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Converting Binary to Decimal
What is the decimal equivalent of the binary
number 010110?
1 x 26
+ 1 x 25
+ 0 x 24
+ 1 x 23
+ 1 x 22
+ 1 x 21
+ 0 x 2º
=
=
=
=
=
=
=
1 x 64 = 64
1 x 32 = 32
0 x 16 = 0
1x8 =8
1x4 =4
1x2 =2
0x1 =0
= 112 in base 10
13
Horner’s Method
Another procedure to calculate the decimal
equivalent of a binary number

Note: This method works with any base
Horner’s Method:





Step 1: Start with the first digit on the left
Step 2: Multiply it by the base
Step 3: Add the next digit
Step 4: Multiply the sum by the base
Step 5: Continue the process until you add the last
digit