Classification of rational numbers and their subsets.

Let’s go step by step to understand what the image is teaching — even though we’re not describing the image, we’ll use its content to explain rational numbers clearly.

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We are learning about rational numbers.

A rational number is any number that can be written as a fraction:
> p / q

Where:
- p and q are integers (like -3, 0, 5, etc.)
- q cannot be zero (because you can’t divide by zero)

Examples from the text:
- 2/3 → yes, rational
- -17/19 → yes, rational
- -3 → also rational! Because it can be written as -3/1
- 0 → rational too! Written as 0/1
- 4.33 → this is 433/100 → still rational!
- Even repeating decimals like 7.123123123... are rational if they repeat in a pattern.

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Now let’s break down the types of rational numbers:

(i) Positive Rational Numbers
If both numerator and denominator are positive OR both are negative → the result is positive.

Example:
5/7 → both positive → positive rational
(-5)/(-7) = 5/7 → same thing → still positive

(ii) Negative Rational Numbers
If one is positive and the other is negative → the result is negative.

Example:
5/(-7) → negative
(-5)/7 → also negative

(iii) Equivalent Rational Numbers
You can multiply or divide both top and bottom by the same non-zero integer → you get an equivalent fraction.

Example:
3/-4 = (-3)/4 = (-6)/8 = (-9)/12 → all equal!

Why? Because multiplying numerator and denominator by the same number doesn’t change the value.

Like:
3/-4 × 2/2 = 6/-8 → which is same as -6/8 → same value!

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Also, there’s a chart showing how numbers are grouped:

All numbers → split into Rational and Irrational
Rational → includes Fractions and Integers
Integers → include Negatives, Zero, Whole Numbers
Whole Numbers → include Zero and Natural Numbers (1, 2, 3...)

So every integer is a rational number — because you can write it over 1.

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Final Answer:
A rational number is any number that can be written as a fraction p/q where p and q are integers and q ≠ 0. This includes positive fractions, negative fractions, whole numbers, zero, and even some decimals. You can create equivalent rational numbers by multiplying or dividing both numerator and denominator by the same non-zero integer.