Number Base System

What is Number Base System?

Number base system is a collection of symbols and rules for representing both small and large numbers.
There are many number systems used today, some are examined below.

Binary Number System

The binary number system is the number system in base 2. There are only two symbols in this system, which are 0 and 1.

Octal Number System

Octal or Oct means eight (8). Hence, the Octal Number System is a number system in base 8. There are eight (8) symbols used in this system. They are: 0, 1, 2, 3, 4, 5, 6, and 7

Decimal or Denary Number System

The decimal Number System is a number system in base ten (10). Ten symbols are used in this system. They include: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

Hexadecimal Number system

Hex and decimal represent 6 and 10 respectively. Therefore, the hexadecimal number system is a number system in base 16. The symbols used are; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. It is important to note that letters A, B, C, D, E and F represent the decimal values of 10, 11, 12, 13, 14 and 15 respectively.

Conversion From One Number Base System To Another

Binary to Decimal Number Conversion

To convert from binary to Decimal number system multiply each binary digit by 2 in an increasing power (right to left) and sum the result.

Example 1
Convert 1101101₂ to a decimal number.
Solution
1101101₂
= (1×2⁶)+(1×2⁵)+(0×2⁴)+1×2³+1×2 ²+0×2¹+1×2⁰

= 1×64+1×32+0×16+1×8+1×4+0×2+1x1
= 64+32+0+8+4+0+1
= 109
Therefore 1101101₂ = 109₁₀

Example 2
Convert 101110011.1101₂ to decimal
Solution
101110011.1101₂
=1×2⁸+0×2⁷+1×2⁶+1×2⁵+1×2⁴+0×2³+0×2²+1×2¹1+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 101110011.1101₂ = 371.40625₁₀

Conversion from Decimal System to Binary System

To convert from a decimal system to a binary number system, Simply continue to divide the given decimal number by two (2) and write down the remainder until it becomes zero (0).
Example 1
Convert 109₁₀ to binary
Solution

2	109	Remainder
2	54	1
2	27	0
2	13	1
2	6	1
2	3	0
2	1	1
2	0	1

Picking the remainder from bottom to top we have 1101101
Therefore, 109 ₁₀ = 1101101₂

Conversion from Octal Base System to Decimal Base System

To convert from an octal base system to a decimal base system, multiply each digit by eight (8) in an increasing power (right to left) of and sum up the result.

Example 1
Convert 3456 base 8 to base ten
Solution
= (3×8³) + (4×8²) + (5×8¹) + (6×8⁰)
= (3×512) + (4×64) + (5×8) + (6×1)
= 1536 + 256 + 40 + 6
= 1838
Therefore 3456₈ = 1838₁₀

Conversion from Decimal to Octal Base System

To convert from a decimal system to an Octal base system, simply divide the given decimal number by eight (8) and write down the remainder until it becomes zero (0).

Example 1
Convert 1838₁₀ to Octal
Solution

8	1838	Remainder
8	229	6
8	28	5
8	3	4
8	0	3
2	3	0

Pick the remainder from bottom to top to get your final answer
Therefore 1838₁₀ = 3456₈

Conversion from Hexadecimal base System to Decimal Base System

To convert from a hexadecimal base system to a decimal base system, multiply each digit by sixteen (16) in an increasing power (right to left) and sum the result.

Example 1
Convert 89F₁₆ to decimal
Solution
89F₁₆
= (8×16²) + (9×16¹) + (F×16⁰)

= (8×16²) + (9×16¹) + (15×16⁰)
= (8×256) + (9×16) + (15×1)
= 2048 + 144 + 15
= 2207
Therefore 89F₁₆ = 2207₁₀

Example 2
Convert CAFE₁₆ to decimal
Solution
CAFE₁₆
= (C×16³) + (A×16²) + (F×16¹) + (E×16⁰)

= (12×16³) + (10×16²) + (15×16¹) + (14×16⁰)
= (12×4096) + (10×256) + (15×16) + (14×1)
= 49152 + 2560 + 240 + 14
= 51966
Therefore CAFE₁₆ = 51966₁₀

Conversion from decimal Base System to hexadecimal Base System

To convert from decimal to hexadecimal base system, continue to divide the given hexadecimal number by sixteen (16) and write down the remainder till it becomes zero (0).
Example 1
Convert 47806₁₀ to Hexadecimal
Solution

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

Picking Remainder in Hex from bottom to top
47806₁₀ = BABE₁₆