Session 4: Types of Numbers | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Session Outline

• A lot of familiar (or less familiar) words about numbers.
• Exploring their precise meaning.
• Building up for problems to come in upcoming sessions.
• Why?
  • Questions may contain these words.
  • Some minor questions we never asked
    • Is 0 a Natural number?
    • Is 19.333333… a rational number?
    • Is π=22/7?

Interesting Sets of Numbers We Talked About

“Precise” Meanings

Natural Numbers: The counting numbers like 1, 2, 3, 4… are called natural numbers.

Whole numbers: Natural numbers along with zero are called whole numbers.

Integers: All counting numbers including zero and the negatives of the counting numbers form the set of integers. Hence …, –3, –2, –1, 0, 1, 2, 3… are all integers.

A few terms to remember: “Positive Integers”, “Negative Integers”, “Non-negative integers” and “Non-positive integers”.

Classification of Integers: Even (divisible by 2), Odd (not divisible by 2).

Rational Numbers (Fractions): A rational number represents a part of a whole or, more generally, any number of equal parts.

Mixed Fraction: An improper fraction can be expressed as a whole number and a proper fraction. This expression is called a mixed fraction.

Decimal representation of rational numbers: When a rational number is written in decimal points, the digits after the decimal will

1. Either stop after a while (Eg. 1.343443)
2. Or repeat the same pattern forever (Eg. 1.546754675467…. “5467” repeats forever).

Irrationals: Numbers with infinite decimal expansion without any pattern are called irrationals. π is an irrational number.

Real numbers: Rational numbers, along with irrational numbers are called real numbers.

Quiz

Additionally, we used terms like, “Sets”, “Venn Diagram”, “[set] contained in [another set]” etc.

Observations from Chair-Arranging Experiment

1. Odd number of chairs cannot be arranged in two rows to make a perfect rectangle.
2. Even number of chairs can be arranged in two rows to make a perfect rectangle.
3. Addition
   • Even + Even = Even
   • Odd + Odd = Even
   • Even + Odd = Odd
   • Odd + Even = Odd
4. Multiplication
   • Even × Even = Even
   • Even × Odd = Even
   • Odd × Even = Even
   • Odd × Odd = Odd

Classification of Natural Numbers

Prime Numbers: A natural number which has exactly two positive factors (divisors), namely itself and 1. For example, 2, 3, 5, 7, 11, …

Composite Numbers: A natural number which has more than two positive factors. For example: 4, 6, 8, 9, 10,…

Note: 1 is neither prime nor composite as it has only one factor. Note the correction from how it was mentioned in the video lecture.

Quizzes

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Here’s a cartoon to end it in a fun note:

More Materials

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