Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems
This assignment is designed to familiarize you with different numbering systems,
specifically: binary, octal, hexadecimal (and decimal) and converting between them.
This lab can be printed out directly or downloaded and printed. Click here for a printable
PDF version of the lab. You can write your responses directly on the printed sheet. To
print this sheet directly: make sure you have a printer available on your machine or
network. In the browser, drag down the File menu and click on print. To download, drag
down your browser’s File menu and select Save as… Then save the file with whatever
name you wish to your machine. You can then print the file.
Note on formatting used below: 1 x 2 ^ 2 equals 1 times 2 raised to 2. The notation
1*2**2 is the same thing. You might see this in a computer language. In the spirit of
being well- rounded both will be used here.
First, a quick review of how to do number conversions.
Decimal: base 10 (0 1 2 3 4 5 6 7 8 9)
Binary: base 2 (0 1)
Octal: base 8 (0 1 2 3 4 5 6 7)
Hexadecimal: base 16 (0 1 2 3 4 5 6 7 8 9 A B C D E F)
Our handy basic number conversion table:
Base 10
base 2
base 8
base 16
Decimal
binary
octal
hex(adecimal)
0
0
0
0
1
1
1
1
2
10
2
2
(base 2 is at 2 digits already)
3
11
3
3
4
100
4
4
5
101
5
5
6
110
6
6
7
111
7
7
8
1000
10
8
(base 8 just went to 2 digits)
9
1001
11
9
------------------------------------------------------------ (base 10 goes to two digits)
10
1010
12
A
11
1011
13
B
12
1100
14
C
13
1101
15
D
14
1110
16
E
15
1111
17
F
(beyond this base 16 goes to 2 digits)
Let’s look at the value of digit “1” at PLACE given the four numbering systems:
POWER
(place)
4
3
2
1
0
Base 10
Base 2
Base 8
Base 16
10000
16
4096
65536
1000
8
512
4096
100
4
64
256
10
2
8
16
1
1
1
1
For example:
POWER
4
(place)
3
2
1
0
Final result
in Decimal
Base 10 of
10001
1 x 10 ^ 4
= 10,000
0 x 10 ^ 3
=0
0 x 10 ^ 2
=
0
0 x 10 ^ 1
=
0
10000 + 0
+0+0+1
= 10001
Base 2 of
10001
1x2 ^4= 0x2^3
16
=
0
0x2^2
=
0
0x 2 ^ 1
=
0
Base 8 of
10001
1x8^4=
4096
0x8^3
=
0
0x8^2
=
0
0x8^1
=
0
Base 16 o
10001
1 x 16 ^ 4
= 65536
0 x 16 ^ 3
=
0
0 x 16 ^ 2
=
0
0 x 16 ^ 1
=
0
1 x 10 ^ 0
=
1x1=
1
1x2^0
=
1x1=
1
1x8^0
=
1x1=
1
1 x 16 ^ 0
=
1x1=
1
16 + 0 + 0
+0+1=
17
4096 + 0 +
0+0+1=
4097
65536 + 0
+0+0+1
= 65537
Going from binary/octal/hexadecimal to decimal:
An easy way to remember it: take the number and raise the base to the place
binary to decimal example
101 = 1*2**2 + 0*2**1 + 1*2**0 = 4 + 0 + 1 = 5
That’s 1 times the base (of 2) raised to the place (2) PLUS 0 times the base (again, 2)
raised to the place (1) PLUS 1 times the base (still 2) raised to the place (0)
octal to decimal example
15 = 1*8**1 + 5*8**0 = 8 + 5 = 13
That’s 1 times the base (of 8) raised to the place (1) PLUS 5 times the base (again, 8)
raised to the place (0)
hex to decimal examples
10 = 1*16**1 + 0*16**0 = 16
That’s 1 times the base (of 16) raised to the place (1) PLUS 0 times the base (yup, 16)
raised to the place (0).
1A = 1*16**1 + 10*16**0 = 26
That’s 1 times the base (of 16) raised to the place (1) PLUS 10 (A is 10 in decimal) times
the base (of 16) raised to the place (0). With hex you have to remember what the decimal
equivalent of the numbers are using our handy chart above.
Going from Decimal to Binary (and octal and hex)
One way is to work with the remainder – continually dividing the initial number and then
resulting numbers by the base until it cannot be divided without fractions. Then take the
remainders, beginning with the first one produced, and write them out from right to left.
Example of decimal to binary:
Take the decimal number 100 to convert to binary.
The decimal number is divided by the base (2) to find the quotient and remainder.
The number 100 / 2 = 50 with remainder of 0
The number 50 / 2 = 25 with remainder of 0
The number 25 / 2 = 12 with remainder of 1
The number 12 / 2 = 6 with remainder of 0
The number 6 / 2 = 3 with remainder of 0
The number 3 / 2 = 1 with remainder of 1
The number 1 / 2 = 0 with remainder of 1
Now from the top take the remainders and write from right to left (or from the bottom up
and write left to right). That gives us the binary number: 1100100
Example of decimal to octal:
Take the decimal number 100 to convert to octal.
The decimal number is divided by the base (8) to find the quotient and remainder.
The number 100 / 8 = 12 with remainder of 4
The number 12 / 8 = 1 with remainder of 4
The number 1 / 8 = 0 with remainder of 1
Now from the top take the remainders and write right to left. That gives us the octal
number: 144
Another easy way for smaller numbers is using the place table.
We know for each place in base 2 we are multiplying by two.
We have:
PLACE
6
5
4
3
2
1
0
Decimal
64
Equivalent
if “1”
present
Total of
64
all these
numbers
added
together =
127
Binary
1
equivalent
(a 6 digit
binary
number)
32
16
8
4
2
1
32
16
8
4
2
1
1
1
1
1
1
1
So, for example, how would you represent 45 decimal in binary?
PLACE
6
Decimal
64
Equivalent if
“1” present
45 = these
numbers added
together
Binary
0
equivalent,
Binary:0101101
5
32
4
16
32
1
0
3
8
2
4
8
4
1
1
1
2
0
1
1
0
1
So, ever wonder why everything in computers is 16, 32, 64, 128, 256, 512, etc? Now it
should be making a lot more sense. The computer, of course, is using binary numbers.
Going from binary to octal/hex and back
(recall our basic conversion table)
Base 10 base 2
base 8
base 16
Decimal binary
octal
hex(adecimal)
0
0
0
0
1
1
1
1
2
10
2
2
3
11
3
3
4
100
4
4
5
101
5
5
6
110
6
6
7
111
7
7
8
1000
10
8
9
1001
11
9
----------------------------------------------------------------10
1010
12
A
11
1011
13
B
12
13
14
15
1100
1101
1110
1111
14
15
16
17
C
D
E
F
Octal is every three bits starting from right- most digit.
Example:
10010011 = 10 010 011 (grouping the digits three at a time from the right- most digit)
NOW, simply find the octal value for each grouping of three in order from left and that is
the final result–
BINARY
OCTAL
10
2
010
2
011
3
10010011
223
…you can do the same thing going backwards to binary.
Hex works the same way. But, it is every 4 bits starting from right-most digit.
Example:
10010011 = 1001 0011
BINARY
OCTAL
1001
9
0011
3
10010011
93
Questions
Ok, time for you to do a little work. Use the number conversion program to check your
answers. You must show your work.
1. Take the following decimal numbers and convert them to binary, octal and
hexadecimal:
20
35
127
245
768
2. Take the following binary numbers and convert to octal, hex, and decimal:
011001
011101
0110
1010101
3. Take the fo llowing octal numbers and convert to binary and hex:
77
35
20
467
4. Take the following hex numbers and convert to binary and octal:
1F
FE
A23
A1B2C3