Numbering systems.
Numbering systems are used for counting. Some numbering systems are commonly used everyday, others
are not as common but do help in understanding the basic fundamentals of numbering systems.
Decimal numbering system
Decimal is the numbering system you all know.
It is also known as base 10.
There are 10 possible digits 0,1,2,3,4,5,6,7,8,9
In a multidigit decimal number, every position has a weighting factor based on the power of 10.
Example:
5
6
7
3(10)
**** the number 5673 can be broken down to induvial digits every incrementing digit also raises the
weighting factor based on power of ten 10^3 = 1000 10^2 = 100 10^1 = 10 10^0 = 1
Base 5 numbering system.
Base 5 is similar to base 10, but only has 5 possible digits 0,1,2,3,4
Due to only having 5 digits, every position has a weighting factor based in the power of 5.
3
2
1(5)
5 being the sub script
1x5^0 2x5^1 3x5^2
To convert the above example, we can add the values together, multiplied by the associated base 5 weight
factor.
(3x5^2)+(2x5^1)+(1x5^0)
75+10+1 = 86(10)
10 being a subscript
Therefore 321(5) = 86(10)
Octal numbering system
-octal is also known as base 8.
-octal has 8 possible digits 0,1,2,3,4,5,6,7
Every position has a weighting factor based on the power of 8
Example
7
5
2
6(8)
8 being the subscript
7x8^3 7x8^2 7x8^1 7x8^0
Lets convert the above example to base 10
7526(8) to base 10
(7x8^3)+(7x8^2)+(7x8^1)+(7x8^0)
3584+320+16+6 = 3926(10)
10 being the subscript
Therefore 7527(8) = 3927(10)
Convert the following the base10
1324(5)
(1x5^3)+(3x5^2)+(2x5^1)+(4x5^0)
125+75+10+4 = 214
Therefore 1324(5) = 214(10)
411(5)
(4x5^2)+(1x5^1)+(1x5^0)
100+5+1= 106(10)
Therefore 411(5) = 106(10)
326(8)
(3x8^2)+(2x8^1)+(6x8^0)
192+16+6= 214(10)
Therefore 326(8) = 214(10)
5126(8)
(5x8^3)+(1x8^2)+(2x8^1)+(7x8^0)
2560+64+16+7=2647(10)
Therefore 5126(8) = 2647(10)
629(8)
(6x8^2)+(2x8^1)+(9x8^0)
384+16+9=409(10)
Therefore 629(8) = 409(10)
This last question is wrong the answer is invalid because it isn’t in the range of 8 numbers
(0,1,2,3,4,5,6,7)
Binary
-Also known as base 2
-Binary has 2 possible digits:0,1
-Every position has a weighting factor based on the power of two
- Binary is used in computers and electronics as a mean to communicate
- different interpretations
0/LOW/OFF/FALSE
1/HIGH/ON/TRUE
-with that said an electrical signal can be converted to binary code by changing states from a positive
voltage (like 5V or in ON State) to ground (like 0V, or an OFF state)
-Binary code is generally written in groupings of 4 or 8 (if not more)
1- Digit is called a bit.
0=>Bit
4- Digits is called a nibble
0110=>Nibble
8- Digits is called a byte.
1011 0011 =>Byte
Lets break down a binary Byte.
1
1
0
0
1
0
0
1(2)
2^7
2^6
2^5
2^4
2^3
2^2
2^1
2^0
128
64
32
16
8
4
2
1
two being a subscript.
To convert the above example to decimal, add up the valued weight of every “1” digit(skip over 0’s)
128+64+8+1 = 201(10) 10 being a subscript
Comparing numbering systems
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Base 5
0
1
2
3
4
10
11
12
13
14
20
21
22
23
Octal
0
1
2
3
4
5
6
7
Octal
0
1
2
3
4
5
6
7
10
11
13
14
15
16
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
Binary
000
001
010
011
100
101
110
111
Hexadecimal(Base16)
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Hexadecimal(base 16)
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
15
F
1111