Number Sets
IMAT Interactive Study Tool
Natural Numbers (\(\mathbb{N}\))
Natural numbers are the positive "counting numbers" starting from 1. They are used to count objects.
\(\mathbb{N} = \{1, 2, 3, 4, ...\}\)
Note: While some definitions include 0, the standard IMAT convention usually starts with 1.
Integers (\(\mathbb{Z}\))
Integers include all natural numbers, their negative opposites, and zero. They represent whole quantities that can be positive, negative, or zero.
\(\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}\)
The set of natural numbers is a subset of the integers (\(\mathbb{N} \subset \mathbb{Z}\)).
Rational Numbers (\(\mathbb{Q}\))
A rational number is any number that can be expressed as a ratio or fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and the denominator \(q\) is not zero.
In decimal form, rational numbers are either terminating (e.g., \(0.5 = \frac{1}{2}\)) or recurring (e.g., \(0.666... = \frac{2}{3}\)).
Irrational Numbers
An irrational number cannot be expressed as a simple fraction. Its decimal representation goes on forever without repeating.
Famous examples include:
- \(\pi \approx 3.14159...\) (The ratio of a circle's circumference to its diameter)
- \(e \approx 2.71828...\) (Euler's number)
- The square root of any non-perfect square, like \(\sqrt{2}\), \(\sqrt{3}\), etc.
1. True or False: Every integer is a rational number.
2. True or False: The number \(\sqrt{9}\) is an irrational number.
Example 1: Classifying Numbers
Classify the following into the smallest number set they belong to (Natural, Integer, Rational, Irrational): -7, \(\sqrt{36}\), 2.5, \(\sqrt{3}\)
Step 1: Analyze each number individually.
- -7: It's a negative whole number. Not Natural, but it is an Integer.
- \(\sqrt{36}\): Simplify first. \(\sqrt{36} = 6\). This is a positive counting number, so it's a Natural number.
- 2.5: A terminating decimal (\(=\frac{5}{2}\)). Not an integer. It's a Rational number.
- \(\sqrt{3}\): 3 is not a perfect square. This is an Irrational number.
Example 2: Converting a Recurring Decimal
Show that \(x = 0.777...\) is rational by converting it to a fraction.
Step 1: Let \(x = 0.777...\)
Step 2: Since one digit repeats, multiply by 10: \(10x = 7.777...\)
Step 3: Subtract the first equation from the second:
\(10x - x = (7.777...) - (0.777...)\)
\(9x = 7\)
Step 4: Solve for x: \(x = \frac{7}{9}\)
Conclusion: Since it can be written as a fraction, it is a rational number.
| Set | Symbol | Definition | Examples |
|---|---|---|---|
| Natural | \(\mathbb{N}\) | Positive counting numbers | 1, 2, 100 |
| Integer | \(\mathbb{Z}\) | Whole numbers (positive, negative, zero) | -5, 0, 8 |
| Rational | \(\mathbb{Q}\) | Can be written as a fraction \(\frac{p}{q}\) | -2, 0.75, \(\frac{1}{3}\) |
| Irrational | N/A | Cannot be written as a fraction | \(\pi\), \(\sqrt{2}\), \(e\) |