Using the ideas discussed in this section, try refining the number system you might have made earlier.

Question

Homi · Accepted Answer

We expand our understanding of numbers to solve more kinds of problems.

Step 1 — Natural Numbers and Whole Numbers

We begin with numbers used for counting. These are called Natural Numbers. Imagine counting your friends or apples.

$\boxed{\text{Natural Numbers (N)} = \{1, 2, 3, \dots \}}$

Sometimes we need to represent having nothing. For this, we add zero. These numbers, including zero, are called Whole Numbers.

$\boxed{\text{Whole Numbers (W)} = \{0, 1, 2, 3, \dots \}}$

Step 2 — Integers

What if you owe money, or the temperature goes below zero? We need negative numbers. We add negative counting numbers to our Whole Numbers. These are called Integers.

$\boxed{\text{Integers (Z)} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots \}}$

Step 3 — Rational Numbers

What if you want to share a pizza equally among three friends? You need fractions. Numbers that can be written as a fraction $\frac{p}{q}$ are called Rational Numbers. Here, $p$ and $q$ are Integers, and $q$ cannot be zero.

$\boxed{\text{Rational Numbers (Q)} = \left\{\frac{p}{q} \mid p, q \in \text{Z}, q \neq 0 \right\}}$

This set includes all Integers, as any Integer $p$ can be written as $\frac{p}{1}$ .

Step 4 — Irrational Numbers

Sometimes, when we measure lengths, like the diagonal of a square, the length cannot be written as a simple fraction. These special numbers are called Irrational Numbers. Examples are $\sqrt{2}$ or $\pi$ . They have decimal forms that never end and never repeat.

$\boxed{\text{Irrational Numbers (I)} = \{\text{Numbers that cannot be written as a fraction}\}}$

Step 5 — Real Numbers

When we combine all Rational Numbers and all Irrational Numbers, we get the Real Numbers. This set covers every number we can place on a number line.

$\boxed{\text{Real Numbers (R)} = \{\text{All Rational and Irrational Numbers}\}}$

Answer

The refined number system includes: (i) Natural Numbers (N): For counting objects. (ii) Whole Numbers (W): Natural Numbers including zero. (iii) Integers (Z): Whole Numbers and their negative counterparts. (iv) Rational Numbers (Q): Numbers that can be expressed as a fraction $\frac{p}{q}$ . (v) Irrational Numbers (I): Numbers that cannot be expressed as a fraction. (vi) Real Numbers (R): All Rational and Irrational Numbers combined.

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