Numbers & Number System

Numbers are mathematical objects used to measure, count, and label. They can be classified into different sets known as number systems. Understanding these systems is fundamental in mathematics. Here are the major categories:

Types of Numbers

Natural Numbers (ℕ)

• Definition: Counting numbers starting from 1 upwards (1, 2, 3, 4, …).
• Usage: Used for counting objects.
• Example: 1, 2, 3, 4, 5,…

Whole Numbers (𝕎)

• Definition: All natural numbers including zero (0, 1, 2, 3, …).
• Usage: Counting with the inclusion of zero.
• Example: 0, 1, 2, 3, 4,…

Integers (ℤ)

• Definition: All whole numbers and their negative counterparts (…, -3, -2, -1, 0, 1, 2, 3, …).
• Usage: Represent quantities with direction (positive or negative).
• Example: -5, 0, 7,…

Rational Numbers (ℚ)

• Definition: Numbers that can be expressed as a fraction p/q, where p and q are integers, and q ≠ 0.
• Properties:
  • Decimal representations are terminating or repeating.
• Example: 1/2, -3/4, 5 (since 5 = 5/1).

Irrational Numbers

• Definition: Numbers that cannot be expressed as a simple fraction of two integers.
• Notation: No standard symbol, but often described as ℝ \ ℚ (real numbers excluding rational numbers).
• Properties:
  • Decimal representations are non-terminating and non-repeating.
• Example: √2, π (pi), e (Euler’s number).

Real Numbers (ℝ)

• Definition: All rational and irrational numbers.
• Usage: Any value along the continuous number line.
• Example: -3, 0, 1/2, √5, π.

Complex Numbers (ℂ)

• Definition: Numbers in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
• Components:
  • Real Part (a): The real component.
  • Imaginary Part (bi): The imaginary component.
• Special Cases:
  • If a = 0, the number is a pure imaginary number.
  • If b = 0, the number is a real number.
  • If both a and b are integers (e.g., 3 + 4i), the number is a Gaussian integer.
• Example: 3 + 2i, -1 – i, 0 + 4i.

Subset Relationships

Each type of number system is a subset of the next:

ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ

Numeral Systems (Number Systems with Different Bases)

Numbers can be represented using different numeral systems based on various bases. Here are the most common ones:

Decimal Number System (Base 10)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Place Value: Each position represents a power of 10.
• Example: The decimal number 123.45 can be expressed as:

1×10²+2×10¹+3×10⁰+4×10⁻¹+5×10⁻² .

Binary Number System (Base 2)

• Digits Used: 0, 1.
• Place Value: Each position represents a power of 2.
• Example: The binary number 1011 can be expressed as:

1×2³+0×2²+1×2¹+1×2⁰ = 8+0+2+1 = 11 in decimal.

Octal Number System (Base 8)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7.
• Place Value: Each position represents a power of 8.
• Example: The octal number 157 can be expressed as:

1×8²+5×8¹+7×8⁰ = 64+40+7 = 111 in decimal.

Hexadecimal Number System (Base 16)

• Digits Used: 0–9 and A–F (where A=10, B=11, …, F=15).
• Place Value: Each position represents a power of 16.
• Example: The hexadecimal number 1A3 can be expressed as:

1×16²+A×16¹+3×16⁰ = 256+160+3 = 419 in decimal.

Converting Between Number Systems

Decimal to Binary Conversion (Example: Convert 13 to Binary)

1. Divide the decimal number by 2.
2. Record the remainder.
3. Continue dividing the quotient by 2 until you reach 0.
4. Read the remainders in reverse order (from bottom to top).
   • Steps
     • Binary Result: Reading remainders bottom to top: 1101.

Binary to Decimal Conversion (Example: Convert 1011 to Decimal)

1. Multiply each binary digit by 2 raised to the power of its position index (starting from 0 on the right).
2. Sum the results.
3. Calculation:

1×2³+0×2²+1×2¹+1×2⁰ = 8+0+2+1 = 111.

Decimal to Hexadecimal Conversion (Example: Convert 255 to Hexadecimal)

1. Divide the decimal number by 16.
2. Record the remainder (use hexadecimal digits for remainders 10–15).
3. Continue dividing the quotient by 16 until you reach 0.
4. Read the remainders in reverse order.
   • Steps:
     • Hexadecimal Result: Reading remainders bottom to top: FF.

Hexadecimal to Decimal Conversion (Example: Convert 1A3 to Decimal)

1. Multiply each hexadecimal digit by 16 raised to the power of its position index.
2. Calculation:

1×16²+10×16¹+3×16⁰ = 256+160+3 = 419.

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