Overview of Number System

Definition of Number System
A number system is a collection of symbols used to represent small numbers, together with a system of rules for representing larger numbers.
There are various number systems, some are examined below:
Decimal Number System
The decimal numeral system (also called base ten or occasionally denary) uses various symbols (digits) for no more than ten distinct values (0, 1, 3, 4, 5, 6, 7, 8, and 9)
Binary Number System
The binary number system is a number system in base 2 and therefore requires only two digits, 0 and 1.
Hexadecimal Number system
The hexadecimal is base 16 digits. The digits 0 through 9 are used, along with the letters A through F, which represent the decimal values 10 through 15.
Octal Number System
The octal numeral system, or oct for short, is the base-8 number system and uses the digits 0 to 7

Conversion From one Number System to Another

Binary Number Conversion

a. Conversion from Binary to Octal Number System
A binary-to-octal table conversion is needed to directly convert from binary to octal. The table is given below. Binary to Octal Table Conversion

Bin	000	001	010	011	100	101	110	111
Oct	0	1	2	3	4	5	6	7

Next, group the binary digits into sets of threes starting with the least significant (rightmost) digits. Then look up each group in the table above.
Example 1: convert 11100101₂ to octal number system.
Solution
Add leading zeros or remove leading zeros to group into sets of three binary digits.
11100101₂ = 011 100 101
By looking up these values in the table above, 011 is 3, 100 is 4 and 101 is 5
Therefore 11100101₂ = 345₈
Method 2
Using normal Mathematics method which involves converting 11100101₂ to decimal number system and thereafter convert the result to an octal number system.
To convert 11100101₂ to decimal multiplying each digits by the base in an increasing power starting with the least significant (rightmost) digit.
11100101₂
= 1x2⁷+1x2⁶+1x2⁵+0x2⁴+0x2 ³+1x2²+0x2¹+1x2⁰
= 128+64+32+0+0+4+0+1
= 229₁₀
Next convert 229₁₀ to base eight by dividing 229 by 8 and writing down the remainder "R"

8	229	R
8	25	5
8	3	4
	0	3

Pick “R” from bottom to top
Therefore, 11100101₂ = 345₈

b. Binary to Hexadecimal
A binary to Hexadecimal table conversion is needed to directly convert from binary to Hex. The table is given below.

Binary to Hexadecimal Table Conversion

Bin	0000	0001	0010	0011	0100	0101	0110	0111
Hex	0	1	2	3	4	5	6	7
Bin	1000	1001	1010	1011	1100	1101	1110	1111
Hex	8	9	A	B	C	D	E	F

Next, group the binary digits into sets of four, starting with the least significant (rightmost) digits.Then, look up each group in the table above.
Example: Convert 11100101₂ to hex
Solution
11100101 = 1110 0101
Looking at the table above 1110 is E and 0101 is 5
Therefore 11100101₂ = E5₁₆

c. Binary to Decimal
There are many methods of converting Binary to decimals. Let’s use the normal Mathematics method which involves multiplying each digits by the base in an increasing power starting with the least significant (rightmost) digit.
Example 1: Convert 111100000000 binary to decimal
Solution
11110000000₂
=(1×2¹⁰)+(1×2⁹)+(1×2⁸)+(1×2⁷)+(1×2⁶)+(1×2⁵)+(1×2⁴)+(1×2³)+(1×2²)+(0×2¹)1+(0×2⁰)
=1024+512+256+128+0+0+0+0+0+0+0
=1920
Therefore 11110000000₂ =1920₁₀

Example 2: Convert 101110011.1101₂ to decimal
Solution
101110011.1101₂
=(1×2⁸)+(0×2⁷)+(1×2⁶)+(1×2⁵)+(1×2⁴)+(0×2³)+(0×2²)+(1×2¹)1+(1×2⁰)+(1×2^-1)+(1×2^-2)+(0×2^-3+(1×2^-4)
=1×256+0×128+1×64+1×32+1×16+0×8+0×4+1×2+1×1+1×1/4+1×1/8+0×1/16+1×1/32
=256+0+64+32+16+0+0+2+1+1/4+1/8+0+1/32
=467+0.25+0.125+0+0.03125
=371.40625
Therefore (101110011.1101)₂ = (371.40625)₁₀

Decimal Number Conversion

a. Decimal to Binary
To convert from Decimal to binary, divide the given decimal number by 2 and write down the remainder until the decimal number becomes zero
Example 1: convert 1792 decimal to binary
Solution

2	1920	R
2	960	0
2	480	0
2	240	0
2	120	0
2	60	0
2	30	0
2	15	0
2	7	1
2	3	1
2	1	1
	0	1

Picking the remainder from bottom to top, we have 11110000000
Therefore 1920₁₀ = 11110000000₂

Example 2: Convert 371.40625₁₀ to binary
Solution
To solve this problem, divide 371 by 2 and put the remainder until it becomes zero

2	371	R
2	185	1
2	46	1
2	23	0
2	11	1
2	5	1
2	2	1
2	1	0
	0	1

Picking the number from bottom to top we have 10111001
371 = 101110011.
Next is to work on the number after the decimal point.
Multiply the decimal number part by 2, write down the whole (W) number part and continue to multiply the decimal result till it becomes 0 (zero)

2	W	.40625
2	0	.8125
2	1	.625
2	1	.25
2	0	.5
	1	.0

Picking "W" from top to bottom, we have 01101 which is the same as 1101
Merging the two results we have 101110011.1101
Therefore 371.40625₁₀ = 101110011.1101₂

b. Decimal to Octal
Example 1: Convert 1792₁₀ to base 8
Solution

8	1792	R
8	224	0
8	28	0
8	3	4
	0	3

Therefore 1792₁₀ = 3400₈

c. Decimal to Hexadecimal
Example 1: Convert 1792 base 10 to hex
Solution

16	1792	R
16	112	0
16	7	0
	0	7

Therefore 1792₁₀ = 700_hex
It is important to note that, converting from decimal to hexadecimal a table must be used to obtain the hexadecimal digit if the remainder is greater than decimal 9.

Relationship between Decimal and Hexadecimal

Dec	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

Example 2: Convert 47806₁₀ to hex
Solution

16	47806	R	Hex R
16	2987	14	E
16	186	11	B
16	11	10	A
	0	11	B

Picking Hex "R" from bottom to top
47806₁₀ = BABE₁₆
Other fun hexadecimal numbers include AD, BE, FAD, FADE, ADD, BED, BEE, BEAD, DEAF, FEE, ODD, BOD, DEAD, DEED, BABE, CAFE, FED, FEED, FACE, BAD

Octal Number Conversion

a. Octal to Binary
Converting from octal to binary is as easy as converting from binary to octal. Simply look up each octal digit to obtain the equivalent group of three binary digits.
Octal to Binary Conversion Table

Bin	000	001	010	011	100	101	110	111
Oct	0	1	2	3	4	5	6	7

Example 1: Convert 345₈ to binary
Solution
From the conversion table
3 =011
4 = 100
5 = 101
Therefore 345₈ = 011100101₂ = 11100101₂

b. Octal to Hexadecimal
When converting from octal to hexadecimal, it is often easier to first convert the octal number into binary and then from binary into hexadecimal.
Example 1: Convert 345₈ to hexadecimal
Solution
Octal = 3 4 5
Binary = 011 100 101
therefore 345₈ = 011100101₂ = 11100101₂
Binary 11100101 = 1110 0101
Then, look up the groups in a table to convert them to hexadecimal digits.

Binary to Hexadecimal Table Conversion

Bin	0000	0001	0010	0011	0100	0101	0110	0111
Hex	0	1	2	3	4	5	6	7
Bin	1000	1001	1010	1011	1100	1101	1110	1111
Hex	8	9	A	B	C	D	E	F

Binary = 1110 0101
Hexadecimal = E 5 = E5_hex Therefore, through a two-step conversion process, octal 345 equals binary 011100101 equals hexadecimal E5.

Octal to Decimal
Example 1: Convert 345₈ to decimal
Solution
Using the usual mathematics method of multiplying each digit by increasing power we have
345 octal = (3 x 8² ) + (4 x 8 ¹) + (5 x 8 ⁰)
= (3 * 64) + (4 * 8) + (5 * 1)
= 229 decimal

Hexadecimal Number Conversion

a. Hexadecimal to Binary
Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.

Binary to Hexadecimal Table Conversion

Bin	0000	0001	0010	0011	0100	0101	0110	0111
Hex	0	1	2	3	4	5	6	7
Bin	1000	1001	1010	1011	1100	1101	1110	1111
Hex	8	9	A	B	C	D	E	F

Example 1: convert A2DE₁₆ to binary
From the conversion table above, we have that
A = 1010
2 = 0010
D = 1101
E = 1110
Putting the binary number together we have
A2DE₁₆ = 1010001011011110₂

b. Hexadecimal to Octal
When converting from hexadecimal to octal, it is often easier to first convert the hexadecimal number into binary and then from binary into octal.
Example 1: Convert A2DE₁₆ to octal
Solution

Binary to Hexadecimal Table Conversion

Bin	0000	0001	0010	0011	0100	0101	0110	0111
Hex	0	1	2	3	4	5	6	7
Bin	1000	1001	1010	1011	1100	1101	1110	1111
Hex:	8	9	A	B	C	D	E	F

From the above table
A = 1010
2 = 0010
D = 1101
E = 1110
Therefore A2DE₁₆ = 1010001011011110₂
Next is to convert 1010001011011110₂ to Octal number system
Add leading zeros or remove leading zeros to group into sets of three binary digits.
1010001011011110₂ = 1 010 001 011 011 110 = 001 010 001 011 011 110
Then, look up each group in a table:

Binary to Octal Table Conversion

Bin	000	001	010	011	100	101	110	111
Oct	0	1	2	3	4	5	6	7

001 = 1
010 = 2
001 = 1
011 = 3
011 = 3
110 = 6
It implies that 1010001011011110₂ = 121336 octal
Therefore, through a two-step conversion process, hexadecimal A2DE equals binary 1010001011011110 equals octal 121336.

c. Hexadecimal to Decimal
Converting hexadecimal to decimal can be performed in the conventional mathematical way, by showing each digit place as an increasing power of 16. Of course, hexadecimal letter values need to be converted to decimal values before performing the math.

Relationship between Decimal and Hexadecimal

Deci	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

Example 1:Convert A2DE hexadecimal to decimal
Solution
A2DE_hex
= ((A) * 16 ³) + (2 * 16 ²) + ((D) * 16 ¹) + ((E) * 16 ⁰)
= (10 * 16 ³) + (2 * 16 ²) + (13 * 16 ¹) + (14 * 16 ⁰)
= (10 * 4096) + (2 * 256) + (13 * 16) + (14 * 1)
= 40960 + 512 + 208 + 14
= 41694 decimal
Therefore, A2DE₁₆ = 41694₁₀