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Number Sets
IMAT Interactive Study Tool
1. Core Theory
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})).
Integers ((mathbb{Z}))
Natural Numbers ((mathbb{N}))
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.
2. Concept Check
1. True or False: Every integer is a rational number.
2. True or False: The number (sqrt{9}) is an irrational number.
3. Solved Examples
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.
4. MCQ Practice
1. Which of the following numbers is irrational?
2. Consider the number line below. The point P is between 2 and 3. Which statement must be false?
2
P
3
3. What is the result of (sqrt{2} times sqrt{8})?
4. Which statement is true?
5. The number 0 is:
5. Summary Table
| 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) |