Overview of Number System

CLASS: SSS Three

Definition of Number System

A number system is a collection of symbols (digits) used to represent numerical values, along with a set of rules for combining these symbols to represent larger numbers. Each number system is defined by its base, which indicates the total number of unique digits (including zero) it uses.

There are various number systems, some of which are examined below:

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Decimal Number System (Base 10)

The Decimal Number System (also called base ten or occasionally denary) is the system we use in our daily lives. It uses ten distinct symbols (digits): 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

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Binary Number System (Base 2)

The Binary Number System is a number system in base 2. It is fundamental to computing and requires only two digits: 0 and 1.

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Hexadecimal Number System (Base 16)

The Hexadecimal Number System is in base 16. It uses digits 0 through 9, along with the letters A through F, which represent the decimal values 10 through 15 respectively.

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

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Octal Number System (Base 8)

The Octal Number System, or "oct" for short, is the base-8 number system. It uses the digits 0 to 7.

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Conversion From One Number System to Another

Binary Number Conversion

a. Conversion from Binary to Octal Number System

To directly convert from binary to octal, a binary-to-octal conversion table is very helpful. The table below shows the equivalent 3-bit binary numbers for each octal digit.

Binary to Octal Conversion Table

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

To perform the conversion, first, group the binary digits into sets of three, starting with the least significant (rightmost) digit. If the leftmost group doesn't have three digits, add leading zeros to complete the group. Then, look up each 3-bit group in the table above to find its octal equivalent.

Example 1: Convert 11100101₂ to an octal number.

Solution:

Add 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
• 101 is 5

Therefore, 11100101₂ = 345₈

Method 2: Conversion via Decimal

Another method involves converting the binary number to the decimal number system first, and then converting the decimal result to an octal number system.

To convert 11100101₂ to decimal, multiply each digit by its respective power of the base (2), starting with 2⁰ for the rightmost digit and increasing the power to the left:

111001012
= (1 × 27) + (1 × 26) + (1 × 25) + (0 × 24) + (0 × 23) + (1 × 22) + (0 × 21) + (1 × 20)
= (1 × 128) + (1 × 64) + (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
= 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) at each step. Read the remainders from bottom to top.

Divide by 8	Quotient	Remainder (R)
8	229
8	28	5
8	3	4
8	0	3

Picking the remainders from bottom to top, we get 345.

Therefore, 11100101₂ = 345₈

b. Binary to Hexadecimal

A binary to hexadecimal conversion table is needed to directly convert from binary to hexadecimal. The table below shows the equivalent 4-bit binary numbers for each hexadecimal digit.

Binary to Hexadecimal Conversion Table

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

Next, group the binary digits into sets of four, starting with the least significant (rightmost) digits. If the leftmost group doesn't have four digits, add leading zeros to complete the group. Then, look up each 4-bit group in the table above to find its hexadecimal equivalent.

Example: Convert 11100101₂ to hexadecimal.

Solution:

111001012 = 1110  0101
    

Looking at the table above:

• 1110 is E
• 0101 is 5

Therefore, 11100101₂ = E5₁₆

c. Binary to Decimal

There are many methods for converting binary numbers to decimals. The most common mathematical method involves multiplying each binary digit by its corresponding power of the base (2), starting with 2⁰ for the least significant (rightmost) digit and increasing the power to the left. Then, sum all the results.

Example 1: Convert 111100000000₂ to decimal.

Solution:

1111000000002
= (1 × 211) + (1 × 210) + (1 × 29) + (1 × 28) + (0 × 27) + (0 × 26) + (0 × 25) + (0 × 24) + (0 × 23) + (0 × 22) + (0 × 21) + (0 × 20)
= (1 × 2048) + (1 × 1024) + (1 × 512) + (1 × 256) + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0
= 2048 + 1024 + 512 + 256
= 3840
    

Therefore, 111100000000₂ = 3840₁₀

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 + 0.8125
= 371.8125
    

Therefore, 101110011.1101₂ = 371.8125₁₀

Decimal Number Conversion

a. Decimal to Binary

To convert a number from decimal (base 10) to binary (base 2), divide the given decimal number by 2 and write down the remainder. Continue this process until the quotient becomes zero (0). The binary equivalent is obtained by picking the remainders from bottom to top.

Example 1: Convert 1920₁₀ to binary.

Solution:

Divide by 2	Quotient	Remainder (R)
2	1920
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
2	0	1

Picking the remainders from bottom to top, we have 111100000000.

Therefore, 1920₁₀ = 111100000000₂

Example 2: Convert 371.8125₁₀ to binary.

Solution:

First, convert the integer part (371) by repeatedly dividing by 2:

Divide by 2	Quotient	Remainder (R)
2	371
2	185	1
2	92	1
2	46	0
2	23	0
2	11	1
2	5	1
2	2	1
2	1	0
2	0	1

Picking the remainders from bottom to top for 371, we get 101110011.

Next, convert the fractional part (0.8125) by repeatedly multiplying by 2 and taking the whole number part:

Multiply by 2	Result	Whole Part (W)
0.8125 × 2	1.625	1
0.625 × 2	1.25	1
0.25 × 2	0.5	0
0.5 × 2	1.0	1

Picking the "Whole Part" from top to bottom for 0.8125, we get .1101.

Merging the two results, we have 101110011.1101.

Therefore, 371.8125₁₀ = 101110011.1101₂

b. Decimal to Octal

To convert from decimal (base 10) to octal (base 8), 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 1792₁₀ to base 8.

Solution:

Divide by 8	Quotient	Remainder (R)
8	1792
8	224	0
8	28	0
8	3	4
8	0	3

Picking the remainders from bottom to top, we get 3400.

Therefore, 1792₁₀ = 3400₈

c. Decimal to Hexadecimal

To convert from decimal (base 10) to hexadecimal (base 16), 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.

It is important to note that when converting from decimal to hexadecimal, a conversion 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 1: Convert 1792₁₀ to hexadecimal.

Solution:

Divide by 16	Quotient	Remainder (R)	Hex Remainder
16	1792
16	112	0	0
16	7	0	0
16	0	7	7

Picking the Hex Remainder from bottom to top, we get 700.

Therefore, 1792₁₀ = 700₁₆

Example 2: 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 Hex Remainder from bottom to top, we get BABE.

Therefore, 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 straightforward. Simply look up each octal digit in a conversion table to obtain its equivalent group of three binary digits. Then, combine these binary groups to form the complete binary number.

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

Putting the binary numbers together, we get:

3458 = 0111001012
    

Leading zeros can be omitted unless they are part of a specific bit group (e.g., for 001). So, this can be simplified to 11100101₂.

Therefore, 345₈ = 11100101₂

b. Octal to Hexadecimal

When converting from octal to hexadecimal, it is often easiest to first convert the octal number into its binary equivalent, and then from that binary number into hexadecimal.

Example 1: Convert 345₈ to hexadecimal.

Solution:

Step 1: Convert Octal to Binary

Octal   =   3   4   5
Binary  = 011 100 101
    

So, 345₈ = 011100101₂.

Step 2: Convert Binary to Hexadecimal

Now, group the binary digits into sets of four, starting from the right (least significant). Add leading zeros if necessary to complete the leftmost group.

Binary: 0111001012
Group into fours from right: 0101 (from right)
                             1110 (next 4)
                             0000 (remaining '0', padded)

Combined groups: 0000 1110 0101
    

Using the Binary to Hexadecimal Conversion Table:

• 0000 = 0
• 1110 = E
• 0101 = 5

Therefore, 011100101₂ = E5₁₆

So, through a two-step conversion process, octal 345₈ equals binary 011100101₂, which equals hexadecimal E5₁₆.

c. Octal to Decimal

Converting octal to decimal can be performed using the conventional mathematical method: multiply each digit by its corresponding power of the base (8), starting with 8⁰ for the rightmost digit and increasing the power to the left. Then, sum all the results.

Example 1: Convert 345₈ to decimal.

Solution:

3458
= (3 × 82) + (4 × 81) + (5 × 80)
= (3 × 64) + (4 × 8) + (5 × 1)
= 192 + 32 + 5
= 22910
    

Therefore, 345₈ = 229₁₀

Hexadecimal Number Conversion

a. Hexadecimal to Binary

Converting from hexadecimal to binary is straightforward and is the reverse of binary to hexadecimal. Simply look up each hexadecimal digit in a conversion table to obtain its equivalent group of four binary digits. Then, combine these 4-bit binary groups to form the complete binary number.

Binary to Hexadecimal Conversion Table

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

Example 1: Convert A2DE₁₆ to binary.

Solution:

From the conversion table above, we have:

• A = 1010
• 2 = 0010
• D = 1101
• E = 1110

Putting the binary numbers together:

A2DE16 = 10100010110111102
    

Therefore, A2DE₁₆ = 1010001011011110₂

b. Hexadecimal to Octal

When converting from hexadecimal to octal, it is often easiest to first convert the hexadecimal number into its binary equivalent, and then from that binary number into octal.

Example 1: Convert A2DE₁₆ to octal.

Solution:

Step 1: Convert Hexadecimal to Binary

Using the Binary to Hexadecimal Conversion Table (provided above):

• A = 1010
• 2 = 0010
• D = 1101
• E = 1110

Therefore, A2DE₁₆ = 1010001011011110₂

Step 2: Convert Binary to Octal

Now, group the binary digits into sets of three, starting from the right (least significant). Add leading zeros if necessary to complete the leftmost group.

Binary: 10100010110111102
Group into threes from right: 110  111  011  010  001  010
Add leading zeros:             001 010 001 011 011 110
    

Using the Binary to Octal Conversion Table (provided earlier in section 'a. Conversion from Binary to Octal Number System'):

• 001 = 1
• 010 = 2
• 001 = 1
• 011 = 3
• 011 = 3
• 110 = 6

It implies that 1010001011011110₂ = 121336₈

Therefore, through a two-step conversion process, hexadecimal A2DE₁₆ equals binary 1010001011011110₂, which equals octal 121336₈.

c. Hexadecimal to Decimal

Converting hexadecimal to decimal can be performed using the conventional mathematical method: multiply each digit by its corresponding power of the base (16), starting with 16⁰ for the rightmost digit and increasing the power to the left. Remember to convert hexadecimal letter values (A-F) to their decimal equivalents (10-15) before performing the multiplication. Then, sum all the results.

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 1: Convert A2DE₁₆ to decimal.

Solution:

A2DE16
= (A × 163) + (2 × 162) + (D × 161) + (E × 160)
= (10 × 163) + (2 × 162) + (13 × 161) + (14 × 160)
= (10 × 4096) + (2 × 256) + (13 × 16) + (14 × 1)
= 40960 + 512 + 208 + 14
= 41694
    

Therefore, A2DE₁₆ = 41694₁₀