Classify numbers

key notes :

Classifying Numbers

Classifying numbers involves understanding the different types of numbers and their properties. Here’s a breakdown of the main types of numbers and how to classify them:

1. Natural Numbers(N)

  • Definition: Numbers that are used for counting and ordering. They start from 1 and go on infinitely.
  • Examples: 1, 2, 3, 4, 5, 6, 7, …

Note: Natural numbers do not include zero or negative numbers.

2. Whole Numbers(w)

  • Definition: Natural numbers plus zero.
  • Examples: 0, 1, 2, 3, 4, 5, 6, …

Note: Whole numbers include zero but not negative numbers or fractions.

3. Integers(I)

  • Definition: All whole numbers and their negative counterparts.
  • Examples: -3, -2, -1, 0, 1, 2, 3, …

Note: Integers include positive numbers, negative numbers, and zero.

4. Rational Numbers(Q)

  • Definition: Numbers that can be expressed as a fraction where both the numerator and denominator are integers, and the denominator is not zero.
  • Examples: 1/2, -3/4, 5, 0.75 (since 0.75 = 3/4)

Note: All integers, fractions, and finite or repeating decimals are rational numbers.

5. Irrational Numbers(Q’)

  • Definition: Numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal parts.
  • Examples: √2, π (pi), e

Note: Irrational numbers have infinite decimal expansions that do not repeat.

6. Real Numbers(R)

  • Definition: All the numbers that can be found on the number line, including both rational and irrational numbers.
  • Examples: -2, 0, 3.14, √5, π

Note: Real numbers include all rational and irrational numbers.

7. Prime Numbers

  • Definition: Natural numbers greater than 1 that have exactly two distinct factors: 1 and themselves.
  • Examples: 2, 3, 5, 7, 11, 13

Note: The number 2 is the only even prime number.

8. Composite Numbers

  • Definition: Natural numbers greater than 1 that have more than two factors.
  • Examples: 4, 6, 8, 9, 10, 12

Note: Composite numbers can be factored into smaller natural numbers.

9. Even Numbers

  • Definition: Numbers divisible by 2.
  • Examples: -4, 0, 2, 6, 8, 10

Note: Even numbers end in 0, 2, 4, 6, or 8.

10. Odd Numbers

  • Definition: Numbers not divisible by 2.
  • Examples: -3, 1, 5, 7, 9

Note: Odd numbers end in 1, 3, 5, 7, or 9.

11. Absolute Value

  • Definition: The distance of a number from zero on the number line, regardless of direction.
  • Examples: |3| = 3, |-5| = 5

Note: The absolute value of a number is always a non-negative number.

Visual Aids

Number Classification Chart:

Type of NumberIncludesExamples
Natural NumbersCounting numbers1, 2, 3, 4, 5, …
Whole NumbersNatural numbers + 00, 1, 2, 3, 4, …
IntegersWhole numbers + negative numbers-3, -2, -1, 0, 1, 2, 3, …
Rational NumbersFractions and whole numbers1/2, -3/4, 5, 0.75
Irrational NumbersNon-repeating, non-terminating decimals√2, π, e
Real NumbersAll rational and irrational numbers-2, 0, 3.14, √5, π
Prime NumbersNumbers with only two factors2, 3, 5, 7, 11, 13
Composite NumbersNumbers with more than two factors4, 6, 8, 9, 10
Even NumbersDivisible by 2-4, 0, 2, 6, 8, 10
Odd NumbersNot divisible by 2-3, 1, 5, 7, 9
Absolute ValueNon-negative distance from zero

Examples and Practice Problems

  1. Classify the Number 15
    • Answer:
      • Natural Number: Yes
      • Whole Number: Yes
      • Integer: Yes
      • Rational Number: Yes (15 = 15/1)
      • Irrational Number: No
      • Real Number: Yes
      • Prime Number: No (15 = 3 × 5)
      • Composite Number: Yes
      • Even Number: No
      • Odd Number: Yes
  2. Classify the Number -8
    • Answer:
      • Natural Number: No
      • Whole Number: No
      • Integer: Yes
      • Rational Number: Yes (-8 = -8/1)
      • Irrational Number: No
      • Real Number: Yes
      • Prime Number: No
      • Composite Number: No
      • Even Number: Yes
      • Odd Number: No
  3. Classify the Number √3
    • Answer:
      • Natural Number: No
      • Whole Number: No
      • Integer: No
      • Rational Number: No
      • Irrational Number: Yes
      • Real Number: Yes
      • Prime Number: No
      • Composite Number: No
      • Even Number: No
      • Odd Number: No

Classifying Numbers Summary

  • Natural Numbers: Counting numbers (1, 2, 3, …).
  • Whole Numbers: Natural numbers plus zero (0, 1, 2, …).
  • Integers: Whole numbers and their negatives (-3, -2, -1, 0, 1, 2, 3, …).
  • Rational Numbers: Can be expressed as a fraction (1/2, 5, 0.75).
  • Irrational Numbers: Cannot be expressed as a fraction (√2, π).
  • Real Numbers: All numbers on the number line (both rational and irrational).
  • Prime Numbers: Natural numbers greater than 1 with only two factors (2, 3, 5).
  • Composite Numbers: Numbers with more than two factors (4, 6, 8).
  • Even Numbers: Divisible by 2 (-4, 0, 2, 6).
  • Odd Numbers: Not divisible by 2 (-3, 1, 5, 7).
  • Absolute Value: Distance from zero (|3| = 3, |-5| = 5).
  • Prime Factorisation: Decomposition into prime factors (30 = 2 × 3 × 5).

Visual Representation

Number Classification Diagram:

let’s practice!