Number Systems & Data Representation - SS1 Digital Technologies Lesson Notes

SS1 Digital Technologies Lesson Notes

Term 1 — Week 6: Number Systems & Data Representation

Introduction to Data Representation

In everyday life, humans count using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This system is known as the Decimal system (Base 10). However, because computer memory circuits are made up of millions of tiny electronic switches that can only exist in two distinct physical states (ON or OFF), digital hardware must process numbers using a different mathematical framework altogether.

To map data accurately within computing, engineers utilize four primary types of number systems. Each system is defined uniquely by its Base (or Radix), which indicates the total count of distinct characters available to represent values.

The Four Core Number Systems:
• Decimal System (Base 10): Uses 10 digits (0–9). The standard human tracking system.
• Binary System (Base 2): Uses 2 digits (0 and 1). The native language of all electronic CPU logic gates.
• Octal System (Base 8): Uses 8 digits (0–7). Used to shorten binary code sequences into more readable sets.
• Hexadecimal System (Base 16): Uses 16 alphanumeric characters (0–9 and letters A, B, C, D, E, F where A=10, B=11, C=12, D=13, E=14, F=15). Extensively used for hardware MAC addresses and web color codes.

Character & Value Equivalency Chart

Below is a structured tracking grid showing how the exact same numeric amounts are expressed across the four distinct base systems:

Decimal (Base 10)	Binary (Base 2)	Octal (Base 8)	Hexadecimal (Base 16)
0	0000	0	0
1	0001	1	1
2	0010	2	2
4	0100	4	4
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
15	1111	15	F

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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

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).

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

Fun Alphanumeric Hack: Other valid hexadecimal vocabulary words include: AD, BE, FAD, FADE, ADD, BED, BEE, BEAD, DEAF, FEE, ODD, BOD, DEAD, DEED, BABE, CAFE, FED, FEED, FACE, and BAD.

Octal Number Conversion

a. Octal to Binary

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. This simplifies to 11100101₂.

Therefore, 345₈ = 11100101₂

b. Octal to Hexadecimal

When converting from octal to hexadecimal, it is 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, octal 345₈ equals binary 011100101₂, which equals hexadecimal E5₁₆.

c. Octal to Decimal

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

Look up each hexadecimal digit in a conversion table to obtain its equivalent group of four binary digits, then combine them seamlessly.

Example 1: Convert A2DE₁₆ to binary.

Solution:

From the conversion table:

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

Putting the binary numbers together:

A2DE16 = 10100010110111102

Therefore, A2DE₁₆ = 1010001011011110₂

b. Hexadecimal to Octal

First convert the hexadecimal number into its binary equivalent, and then group that binary number into sets of three to establish octal positioning.

Example 1: Convert A2DE₁₆ to octal.

Solution:

Step 1: Convert Hexadecimal to Binary

A2DE16 = 10100010110111102

Step 2: Convert Binary to Octal

Group the binary digits into sets of three, starting from the right (least significant).

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

Using our Binary to Octal Conversion values:

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

Therefore, hexadecimal A2DE₁₆ equals octal 121336₈.

Test Your Knowledge (Week 6 Quiz)

Select the correct answer choice and click 'Submit' to process your conversion score calculation instantly.

1. What is the base or radix value used in the hexadecimal number system?

A) Base 2 B) Base 8 C) Base 10 D) Base 16

2. In the hexadecimal system, what alphanumeric letter character is used to represent the decimal number value 12?

A) A B) B C) C D) F

3. Convert the decimal integer value 9 into its direct 4-bit binary representation string:

A) 0111 B) 1001 C) 1010 D) 1111

4. Translate the binary sequence value 101₂ into its everyday decimal value equivalent:

A) 3 B) 4 C) 5 D) 7

5. Which numbering framework relies entirely on symbols 0, 1, 2, 3, 4, 5, 6, and 7?

A) Binary System B) Octal System C) Hexadecimal System D) Decimal System

6. Why do computer systems use Hexadecimal notation if processors can only read binary code natively?

A) Hexadecimal makes the computer calculate values much faster. B) Hexadecimal acts as a shorter, human-readable shorthand for long binary strings. C) Hexadecimal consumes less RAM storage space than binary numbers. D) Hexadecimal can represent fractional values while binary cannot.

7. Using the bit-grouping shortcut method, how many individual binary bits are precisely represented by a single Octal (Base 8) digit?

A) 2 bits B) 3 bits C) 4 bits D) 8 bits

8. Which of the following numbers represents a completely INVALID entry within an Octal system compilation?

A) 147₈ B) 205₈ C) 768₈ D) 310₈

Class Test & Calculation Assignments

Solve these math problems clearly inside your notebooks showing all breakdown steps:

1. Convert the following decimal integers completely into binary format strings: 14₁₀ and 42₁₀.
2. Convert these binary patterns cleanly back into Base 10 decimal digits: 1110₂ and 10011₂.
3. Explain clearly why values 10 through 15 inside the Hexadecimal system must be written as letters A through F instead of standard numbers.