IT Essentials Lecture 3: 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)

Decimal Number System (Base-10)

• Digits Used: 0~9
• Structure: Each digit represents a power of 10.
• Example: Decimal 527
• Usage: Standard system used in everyday life for counting and arithmetic.

Decimal to Binary Conversion Example

To convert a decimal number to binary:

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

Example: Convert Decimal 13 to Binary

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

Binary: 1101

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.
• Example: Convert Binary 101 to Decimal.
• Usage: Binary is essential for computers, as digital circuits operate in two states (on/off).

Binary Positional Notation

• Definition: Positional notation means each digit represents a different value depending on its position.
• Example Calculation:
  • Binary Number: 1100 0000
  • Calculation: $(1\times 2^8)+(1\times 2^7)+0+0+0+0+0+0=192$

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.

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.

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: 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

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.

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.

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.