Number Base System

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

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Types of Number Base Systems

Binary Number System

The Binary Number System is the number system in base 2. There are only two symbols (digits) 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: 0, 1, 2, 3, 4, 5, 6, and 7.

Decimal or Denary Number System

Decimal Number System is base ten (10). Ten symbols are used in this system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the system we use for everyday counting.

Hexadecimal Number system

Hex and decimal represent 6 and 10 respectively, making the Hexadecimal Number System a base 16 system. The sixteen 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.

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Conversion From One Number Base System To Another

Binary to Decimal Number Conversion

To convert from binary to Decimal, multiply each binary digit by an increasing power (right to left) of 2 and sum the results.

Example 1: Convert 1101101₂ to a decimal number.
Solution
1101101₂
= (1×2⁶)+(1×2⁵)+(0×2⁴)+(1×2³)+(1×2²)+(0×2¹)+(1×2⁰)

= 1×64+1×32+0×16+1×8+1×4+0×2+1×1
= 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+1×2¹+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×&frac{1}{2}+1×&frac{1}{4}+0×&frac{1}{8}+1×&frac{1}{16}
= 256+0+64+32+16+0+0+2+1+0.5+0.25+0+0.0625
= 371 + 0.8125
= 371.8125
Therefore, 101110011.1101₂ = 371.8125₁₀

Conversion from Decimal System to Binary System

To convert from decimal to binary, repeatedly divide the decimal number by two (2) and write down the remainder. The binary result is the remainders read from bottom to top.

Example 1: Convert 109₁₀ to binary.
Solution

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

Picking the remainder from bottom to top: 1101101

Therefore, 109₁₀ = 1101101₂

Conversion from Octal Base System to Decimal Base System

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

Example 1: Convert 3456₈ to base ten.
Solution
3456₈
= (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 decimal to octal, repeatedly divide the decimal number by eight (8) and write down the remainder.

Example 1: Convert 1838₁₀ to Octal.
Solution

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

Picking the remainder from bottom to top: 3456

Therefore, 1838₁₀ = 3456₈

Conversion from Hexadecimal base System to Decimal Base System

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

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

= (12×4096) + (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, repeatedly divide the decimal number by sixteen (16) and write down the remainder.

Example 1: Convert 47806₁₀ to Hexadecimal.
Solution

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

Picking Remainder in Hex from bottom to top: BABE

Therefore, 47806₁₀ = BABE₁₆

Click here for an extensive coverage on number base system

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🧠 Test Your Knowledge 📝

See how much you've learned about Number Base Systems!

1. What is the base of the Hexadecimal Number System?

A. 8 B. 2 C. 10 D. 16

2. Which decimal value is represented by the hexadecimal symbol 'A'?

A. 9 B. 10 C. 11 D. 15

3. What is the binary equivalent of the decimal number 5₁₀?

A. 11₂ B. 100₂ C. 101₂ D. 110₂

4. How many unique symbols (digits) are used in the Octal Number System?

A. 2 B. 8 C. 10 D. 16