Digital Number Systems

Introduction to Numbering Systems

Definition: A numbering system is a way to represent numbers in different bases.

Common Systems:

• Binary (Base-2)
• Octal (Base-8)
• Decimal (Base-10)
• Hexadecimal (Base-16)

Binary Number System (Base-2)

• Digits Used: 0, 1
• Structure: Each digit in a binary number represents a power of 2, with the rightmost digit representing $2^0$, the next $2^1$, and so on.
• Usage: Binary is essential for computers, as digital circuits operate in two states (on/off).

A Bit represents a single digit, Byte represents 8-bits.

• The most significant bit (MSB) is the leftmost bit (largest weight).
• The least significant bit (LSB) is the rightmost bit (smallest weight).

To calculate the largest number that can be represented using N bits, use the following formula:

$$2^N-1$$

Example: What is the largest number that can be represented using eight bits?

$$2^8-1={255}_{10}=1111{1111}_2$$

Summary: Using $N$ bits, we can count through $2^N$ different decimal numbers ranging from 0 to $2^{N-1}$ .

Decimal Number System (Base-10)

• Digits Used: 0~9
• Structure: Each digit represents a power of 10.
• Usage: Standard system used in everyday life for counting and arithmetic.

Hexadecimal Number System (Base-16)

• Digits Used: 0~9, A=10, B=11, C=12, D=13, E=14, F=15
• Structure: Each digit represents a power of 16.
• Usage: Widely used in programming, particularly for color codes in web design (e.g., #FF5733) and memory addresses.

Counting in Hexadecimal

Hexadecimal counting is base-16, where each digit increases from 0 to F. After F, the digit resets to 0 and a carry is added to the next higher position, exactly like decimal counting but with 16 symbols.

Example: 3D, 3E, 3F, 40, 41, 42 (F rolls over to 0 and increments the next digit: 3-> 4).

Using $N$ hex digits, we can count through ${16}^N$ different decimal numbers ranging from 0 to ${16}^{N-1}$ .

Example: What is the largest number that can be represented using three digits?

We can count from $(000{)}_{16}$ to $(FFF{)}_{16}$

$${16}^3-1={4095}_{10}=FF{F}_{16}$$

Octal Number System (Base-8)

• Digits Used: 0~7
• Structure: Each digit represents a power of 8.
• Historical Context: Used as a shorthand for binary in computing, though hexadecimal is more common today.

Conversions Between Number Systems

Binary to Decimal Conversion

1. Each binary digit represents a power of 2.
2. Multiply each binary digit by its positional power of 2 and add the results.

Example: $(1011.101{)}_2=(??{)}_{10}$

Answer: $(1011.101{)}_2=(11.625{)}_{10}$

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Example: $(11011{)}_2=(??{)}_{10}$

Answer: $(11011{)}_2=(27{)}_{10}$

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Decimal to Binary Conversion: Division Method

To convert a decimal number to binary:

1. Divide the number by 2.
2. Write down the quotient (division result) and remainder.
3. Repeat with the quotient until it reaches 0.
4. The binary number is the remainders read in reverse order.

Example 1: $(25{)}_{10}=(??{)}_2$

Answer 1: $(25{)}_{10}=(11001{)}_2$

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Example 2: $(13{)}_{10}=(??{)}_2$

• $13÷2=6$ remainder 1
• $6÷2=3$ remainder 0
• $3÷2=1$ remainder 1
• $1÷2=0$ remainder 1

Answer 2: $(13{)}_{10}=(1101{)}_2$

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Decimal to Binary Conversion: Subtraction Method

1. Start with the largest power of 2 less than or equal to the number.
2. Subtract that power of 2, write a 1 in its position.
3. If the power of 2 is greater than the remainder, write 0.

Example 1: Convert Decimal 213 to Binary:

Powers of 2 to consider:
128 64 32 16 8 4 2 1

Step-by-step:

• (213 - 128 = 85) -> 1 in the 128 position
• (85 - 64 = 21) -> 1 in the 64 position
• 32 > 21 -> 0 in the 32 position
• (21 - 16 = 5) -> 1 in the 16 position
• 8 > 5 -> 0 in the 8 position
• (5 - 4 = 1) -> 1 in the 4 position
• 2 > 1 -> 0 in the 2 position
• (1 - 1 = 0) -> 1 in the 1 position

Binary:
1101 0101

Example 2: Convert Decimal 168 to Binary:

• $168-128=40$ -> 1 in 128 position.
• $40-32=8$ -> 1 in 32 position.
• $8-8=0$ -> 1 in 8 position.

Binary: 1010 1000

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Decimal to Octal Conversion

1. Divide the decimal number by 8.
2. Write down the remainder.
3. Repeat until the quotient is 0.
4. Read the remainders in reverse order.

Example: Convert Decimal 65 to Octal:

• $65÷8=8$ Remainder 1
• $8÷8=1$ Remainder 0
• $1÷8=0$ Remainder 1

Octal: 101

Decimal to Hexadecimal Conversion

1. Divide by 16.
2. Record quotient and remainder.
3. Continue dividing until quotient is 0.
4. Convert remainders above 9 to hexadecimal digits (e.g., 10 = A).

Example: Decimal 255 to Hexadecimal:

• $255÷16=15$ Remainder 15 (F)
• $15÷16=0$ Remainder 15 (F)

Hexadecimal: FF

Octal to Binary Conversion

1. Convert each octal digit to its 3-digit binary equivalent. Use the following reference table:

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

Example: Octal 57 -> Binary 101 111.

Binary to Octal Conversion

1. Group binary digits into sets of three from the right.
2. Convert each group to its octal equivalent.

Example: Binary 110 101 011 -> Octal 653.

Hexadecimal to Binary Conversion

1. Convert each hexadecimal digit to its 4-bit binary equivalent. Use the following reference table:

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

Example: Hexadecimal 2A3 -> Binary 0010 1010 0011.

Binary to Hexadecimal Conversion

1. Group binary digits into sets of four.
2. Convert each group to hexadecimal.

Example: Binary 0001 1010 1101 -> Hexadecimal 1AD.

Summary

• Binary: A base-2 system using 0 and 1, essential for computing.
• Decimal: Base-10 system, standard in everyday life.
• Hexadecimal: Base-16, using 0~9 and A~F, used in IPv6 and MAC addresses.
• Conversions:
  • Binary <-> Decimal <-> Hexadecimal: Often requires intermediate binary representation.
  • Decimal <-> Octal <-> Hexadecimal: Follow division methods or positional notation.

Binary-Coded Decimal (BCD)

BCD represents each decimal digit (0-9) with a 4-bit binary code.

Decimal	BCD (4-bit Binary)
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001

BCD encodes decimal numbers for applications where decimal representation is important, such as digital displays, timers, calculators, and avometers.

Example:

The decimal number 29 is represented in BCD as:

• $2$ = $0010$
• $9$ = $1001$

Therefore, $29$ in BCD is $00101001$.

Example:

Convert $0110100000111001$ (BCD) to its decimal equivalent.

Solution:

Split the number into 4-bit groups:

• $0110$ -> 6
• $1000$ -> 8
• $0011$ -> 3
• $1001$ -> 9

Combine the digits:
Decimal = 6839

BCD vs. Binary

Aspect	Binary Code	BCD Code
Representation of ${137}_{10}$	$1000{1001}_2$	$00010011{0111}_2$
Encoding method	Entire number converted directly to base-2	Each decimal digit encoded separately in 4 bits
Conversion to/from decimal	More complex	Simple and straightforward
Digits to remember	Full binary weighting	Only 4-bit codes for digits 0–9
Hardware implementation	Requires more complex logic	Easier logic circuits
Main advantage	Compact representation	Ease of decimal conversion

Gray Code

Gray code is a binary numbering system where only one bit changes between successive values, reducing errors caused by rapid signal changes in digital systems.

Key application:

Used in shaft position encoders to accurately represent mechanical rotation without ambiguity during transitions.

Conversion Examples

Example 1: Binary -> Gray

Convert Binary 101101 to Gray code.

Rule:

• Gray MSB = Binary MSB
• Each next Gray bit = XOR of current binary bit with previous binary bit

Step-by-step:

Binary (B)	Operation	Gray (G)
1	MSB copied	1
0	$1\oplus 0$	1
1	$0\oplus 1$	1
1	$1\oplus 1$	0
0	$1\oplus 0$	1
1	$0\oplus 1$	1

Result:
Binary 101101 -> Gray 111011

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Example 2: Gray -> Binary

Convert Gray 111011 back to binary.

Rule:

• Binary MSB = Gray MSB
• Each next Binary bit = XOR of current Gray bit with previous binary bit

Step-by-step:

Gray (G)	Operation	Binary (B)
1	MSB copied	1
1	$1\oplus 1$	0
1	$1\oplus 0$	1
0	$0\oplus 1$	1
1	$1\oplus 1$	0
1	$1\oplus 0$	1

Result:
Gray 111011 -> Binary 101101

Byte, Nibble, and Word

• A byte is 8 bits and is the standard unit for storing data.
• A nibble is 4 bits (half a byte).
• A word is the number of bits a system processes at once and depends on the CPU architecture.
  Example:
  A 64-bit processor has a word size of 64 bits (8 bytes), while a simple microcontroller may have an 8-bit word size.

Alphanumeric Codes and ASCII Code

• Computers use alphanumeric codes to represent letters, numbers, punctuation, and symbols.

• These codes allow non-numerical information to be processed and transmitted.

• ASCII is a 7-bit code with 128 possible characters.

• Used for keyboard input, internal storage, and communication with external devices.
  Example:
  ASCII for 'A' = 0100 0001 (decimal 65).

Parity Method for Error Detection

• A parity bit is added to transmitted data to detect errors.
• Even parity: total number of 1s (including parity bit) is even.
• Odd parity: total number of 1s is odd.
  Example:
  Data 101 1001 has four 1s -> even parity bit = 0 (they are already even -> 4).
  Data 110 0011 has four 1s -> odd parity bit = 1 (to make them odd -> 5).

ASCII with Even Parity Example (HELLO)

7-bit ASCII + 1 parity bit:

• H -> 100 1000 (2 ones) -> parity 0 -> 0100 1000
• E -> 100 0101 (3 ones) -> parity 1 -> 1100 0101
• L -> 100 1100 (3 ones) -> parity 1 -> 1100 1100
• L -> 100 1100 (3 ones) -> parity 1 -> 1100 1100
• O -> 100 1111 (5 ones) -> parity 1 -> 1100 1111

These 8-bit strings are what would be transmitted using ASCII with even parity.