Number Base System

TOPIC: Number Base System

CLASS: JSS TWO

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What is a Number Base System?

A number base system is a collection of symbols (digits) and rules used to represent both small and large numbers. It defines how quantities are counted and written. Each number system is characterized by its base, which indicates how many unique digits it uses, including zero.

There are many number systems used today, some of the most common ones are examined below:

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Binary Number System (Base 2)

The Binary Number System is the number system in base 2. It is fundamental to computers because it uses only two symbols (digits): 0 and 1. These correspond to the "off" and "on" states of electronic circuits.

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Octal Number System (Base 8)

The term "Octal" or "Oct" means eight (8). Hence, the Octal Number System is a number system in base 8. It uses eight unique symbols (digits): 0, 1, 2, 3, 4, 5, 6, and 7.

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Decimal or Denary Number System (Base 10)

The Decimal Number System, also known as the Denary Number System, is the number system in base ten (10). This is the system we use every day. It uses ten symbols (digits): 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

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Hexadecimal Number System (Base 16)

"Hex" represents 6 and "decimal" represents 10, so "Hexadecimal" means a total of 16. Therefore, the Hexadecimal Number System is a number system in base 16. The symbols used are:

• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Letters: A, B, C, D, E, and F

It is important to note that the letters represent the following decimal values:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

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Conversion From One Number Base System To Another

Binary to Decimal Number Conversion

To convert a number from binary (base 2) to decimal (base 10), you multiply each binary digit by a power of 2, starting from for the rightmost digit and increasing the power by one for each digit to the left. Then, you sum up all 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)
NOTE: From Laws of Indices, "Any number raised to the power of zero (0) is one (1)".
= (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) + (1×0.5) + (1×0.25) + (0×0.125) + (1×0.0625)
= 256 + 0 + 64 + 32 + 16 + 0 + 0 + 2 + 1 + 0.5 + 0.25 + 0 + 0.0625
= 371 + 0.5 + 0.25 + 0.0625
= 371.8125
    

Therefore, 101110011.1101₂ = 371.8125₁₀

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Conversion from Decimal System to Binary System

To convert a number from a decimal (base 10) to a binary (base 2) number system, simply continue to divide the given decimal number by two (2) and write down the remainder. Collect the remainders from bottom to top to get the binary equivalent.

Example 1:

Convert 109₁₀ to binary.

Solution:

Divide by 2	Quotient	Remainder
2	109
2	54	1
2	27	0
2	13	1
2	6	1
2	3	0
2	1	1
2	0	1

Picking the remainders from bottom to top, we have 1101101.

Therefore, 109₁₀ = 1101101₂

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Conversion from Octal Base System to Decimal Base System

To convert a number from an octal (base 8) to a decimal (base 10) base system, multiply each octal digit by a power of 8, starting from 8⁰ for the rightmost digit and increasing the power by one for each digit to the left. Then, sum up the results.

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

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Conversion from Decimal to Octal Base System

To convert a number from a decimal (base 10) to an octal (base 8) base system, simply divide the given decimal number by eight (8) and write down the remainder. Continue this process until the quotient becomes zero (0). Collect the remainders from bottom to top to get the octal equivalent.

Example 1:

Convert 1838₁₀ to Octal.

Solution:

Divide by 8	Quotient	Remainder
8	1838
8	229	6
8	28	5
8	3	4
8	0	3

Picking the remainders from bottom to top, we get 3456.

Therefore, 1838₁₀ = 3456₈

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Conversion from Hexadecimal Base System to Decimal Base System

To convert a number from a hexadecimal (base 16) to a decimal (base 10) base system, multiply each hexadecimal digit by a power of 16, starting from 16⁰ for the rightmost digit and increasing the power by one for each digit to the left. Remember to substitute hexadecimal letters (A-F) with their decimal equivalents (10-15) before multiplying. Then, sum up the results.

Example 1:

Convert 89F₁₆ to decimal.

Solution:

89F16
= (8 × 162) + (9 × 161) + (F × 160)
NOTE: F = 15
= (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)
NOTE: C = 12, A = 10, F = 15, E = 14
= (12 × 163) + (10 × 162) + (15 × 161) + (14 × 160)
= (12 × 4096) + (10 × 256) + (15 × 16) + (14 × 1)
= 49152 + 2560 + 240 + 14
= 51966
    

Therefore, CAFE₁₆ = 51966₁₀

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Conversion from Decimal Base System to Hexadecimal Base System

To convert a number from decimal (base 10) to hexadecimal (base 16) base system, continue to divide the given decimal number by sixteen (16) and write down the remainder. If the remainder is 10 or greater, convert it to its corresponding hexadecimal letter (A-F). Continue this process until the quotient becomes zero (0). Collect the remainders (in hexadecimal form) from bottom to top to get the hexadecimal equivalent.

Example 1:

Convert 47806₁₀ to Hexadecimal.

Solution:

Divide by 16	Quotient	Remainder (Decimal)	Remainder (Hex)
16	47806
16	2987	14	E
16	186	11	B
16	11	10	A
16	0	11	B

Picking the remainders in Hex from bottom to top, we get BABE.

Therefore, 47806₁₀ = BABE₁₆

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Find SSS 3 notes on the overview of number system here

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