Number Conversions

Absolutely! Here’s a clear explanation that first covers the logic behind each type of conversion, then follows with examples. It’s simple, step-by-step, and easy to understand:

Number Conversions: Logic and Examples

1. Converting Binary to Denary (Decimal)

Logic:

Each digit in a binary number represents a power of 2, starting from

2^{0} = 1

at the rightmost bit.

To find the decimal value, multiply each binary digit by its corresponding power of 2.

Add up all the values where the binary digit is 1.

Example: Convert 11101110 to decimal

Place Value (powers of 2)	128	64	32	16	8	4	2	1
Binary Digit	1	1	1	0	1	1	1	0

Add the place values with 1s:
128 + 64 + 32 + 8 + 4 + 2 = 238

2. Converting Denary to Binary

Logic:

Divide the decimal number by 2 repeatedly.

At each division, note the remainder (0 or 1).

Continue until the quotient is 0.

The binary number is the remainders read from bottom to top.

Example: Convert 142 to binary

Divide by 2	Quotient	Remainder
142 ÷ 2	71	0
71 ÷ 2	35	1
35 ÷ 2	17	1
17 ÷ 2	8	1
8 ÷ 2	4	0
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Read remainders bottom to top → 10001110

3. Converting Hexadecimal to Binary

Logic:

Each hex digit represents a value from 0 to 15.

Convert each hex digit to its 4-bit binary equivalent.

Join all these binary groups together for the full binary number.

Example: Convert 21FD (hex) to binary

Hex Digit	2	1	F	D
Decimal	2	1	15	13
Binary	0010	0001	1111	1101

Final binary: 0010000111111101

4. Converting Binary to Hexadecimal

Logic:

Split the binary number into groups of 4 bits, starting from the right.

Add zeros on the left if the first group has less than 4 bits.

Convert each 4-bit group to decimal, then to hex (0–9 or A–F).

Example: Convert 1000011111101 to hex

Binary Group	0010	0001	1111	1101
Decimal	2	1	15	13
Hex	2	1	F	D

Final hex: 21FD

5. Converting Hexadecimal to Denary

Logic:

Convert hex to binary (as in step 3).

Convert that binary number to decimal (as in step 1).

6. Converting Denary to Hexadecimal

Logic:

Convert decimal to binary (as in step 2).

Convert binary to hex (as in step 4).

Sure! Here are detailed step-by-step solutions for each practice question:

Practice Questions & Worked Examples

1. Binary to Denary (Decimal)

Question: Convert binary 10110101 to decimal.

Step 1: Write place values (powers of 2):

128	64	32	16	8	4	2	1
1	0	1	1	0	1	0	1

Step 2: Add place values where bit = 1:

128 + 32 + 16 + 4 + 1 = 181

2. Denary to Binary

Question: Convert decimal 78 to binary.

Step 1: Divide by 2 repeatedly, write remainders:

Divide by 2	Quotient	Remainder
78 ÷ 2	39	0
39 ÷ 2	19	1
19 ÷ 2	9	1
9 ÷ 2	4	1
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Step 2: Read remainders from bottom to top → 1001110

3. Hexadecimal to Binary

Question: Convert hex 3A9F to binary.

Hex Digit	3	A	9	F
Decimal	3	10	9	15
Binary	0011	1010	1001	1111

Final binary: 0011101010011111

4. Binary to Hexadecimal

Question: Convert binary 11011100101 to hexadecimal.

Step 1: Group bits into 4 from right (add leading zeros if needed):

Binary: 0001 1011 1001 0101

Group	0001	1011	1001	0101
Decimal	1	11	9	5
Hex	1	B	9	5

Final hex: 1B95

5. Hexadecimal to Denary

Question: Convert hex 4B7 to decimal.

Step 1: Convert each hex digit to binary:

Hex Digit	4	B	7
Decimal	4	11	7
Binary	0100	1011	0111

Binary number: 010010110111

Step 2: Convert binary to decimal:

Place Value	2048	1024	512	256	128	64	32	16	8	4	2	1
Binary Digit	0	1	0	0	1	0	1	1	0	1	1	1

Add values with 1: 1024 + 128 + 32 + 16 + 4 + 2 + 1 = 1207

6. Denary to Hexadecimal

Question: Convert decimal 255 to hexadecimal.

Step 1: Convert 255 to binary:

Divide by 2	Quotient	Remainder
255 ÷ 2	127	1
127 ÷ 2	63	1
63 ÷ 2	31	1
31 ÷ 2	15	1
15 ÷ 2	7	1
7 ÷ 2	3	1
3 ÷ 2	1	1
1 ÷ 2	0	1

Binary: 11111111

Step 2: Group binary into 4 bits:

1111 1111

Group	1111	1111
Dec	15	15
Hex	F	F

Final hex: FF