When you think about numbers, you might first think of 1, 2, 3, or even fractions like ½. But did you know there are numbers that cannot be written as simple fractions? These are called irrational numbers, and today, we’ll explore them in a way that’s easy and fun!
What are Irrational Numbers?
An irrational number is a number that cannot be written as a fraction (like p/q, where p and q are integers, and q is not zero). In simple words, if you can’t express a number as a neat fraction, it’s irrational!
Irrational numbers are special because their decimal forms never end and never repeat. They go on forever without forming a pattern.
Simple Definition:
A number that cannot be written as a fraction is called an irrational number.
Examples of Irrational Numbers
Here are some famous irrational numbers you may have heard of:
- √2 = 1.4142135… (goes on forever!)
- π (pi) = 3.1415926… (used in circles!)
- √3, √5, and even numbers like 0.10110111011110… (a pattern that never repeats)
Notice how these numbers never stop and never repeat? That’s what makes them irrational!
Rational Numbers vs. Irrational Numbers
Let’s quickly compare rational and irrational numbers:
| Rational Numbers | Irrational Numbers |
|---|---|
| Can be written as fractions (p/q) | Cannot be written as fractions |
| Decimal ends or repeats | Decimal never ends or repeats |
| Examples: 1/2, 5, -3, 0.75 | Examples: √2, π, √5 |
Rational numbers include numbers like 1, 2, 0.5, and -7. You can write them neatly as a fraction.
Irrational numbers are messy in a beautiful way. They just keep going without a pattern.
Fun Fact! 🌟
The ancient Greeks were shocked when they discovered irrational numbers! The mathematician Hippasus found out that √2 could not be written as a fraction, and it changed how people thought about math forever.
Importance of Irrational Numbers
- They help us understand real-world things like the length of the diagonal of a square or the size of a circle.
- Together with rational numbers, they complete the real numbers, meaning every point on the number line represents either a rational or an irrational number.
Quick Quiz! ✅
Try to guess if these numbers are rational or irrational:
- 4.33333… (Repeats)
- √7
- 0.25
- π
- 1.101001000100001…
Answers:
- Rational
- Irrational
- Rational
- Irrational
- Irrational
NCERT Solutions for Exercise 1.2 – Number System Class 9
In this exercise, we are learning about irrational numbers as part of our study of the number system class 9. Let’s go through each question one by one!
1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
✅ True
Every irrational number lies on the number line, so it is a real number.
(ii) Every point on the number line is of the form √m, where m is a natural number.
❌ False
Not every point is of the form √m. For example, 1/2 is a point on the number line but not of the form √m.
(iii) Every real number is an irrational number.
❌ False
Real numbers include both rational and irrational numbers.
2. Are the square roots of all positive integers irrational? If not, give an example.
Answer: No.
Not all square roots of positive integers are irrational.
📌 Example: √4 = 2 is rational
3. Show how √5 can be represented on the number line.
Answer:
Here’s how you can represent √5 on the number line:
Steps:
- Draw a number line and mark point O as 0.
- Mark 2 units to the right of O, and label the point as A.
- At A, draw a line perpendicular to the number line and mark a distance of 1 unit upwards. Label the endpoint as B.
- Join OB. By Pythagoras theorem, OB = √(2² + 1²) = √5.
- Now, with O as center and OB as radius, draw an arc cutting the number line at point P.
- Point P represents √5 on the number line.
(Tip: In number system class 9, representing irrational numbers on the number line using Pythagoras theorem is a key method.)
Irrational numbers may sound tricky, but they’re just numbers that cannot be neatly expressed as a fraction. Their decimals go on forever without repeating. They are an important part of the number family, standing right alongside rational numbers to form the set of real numbers.
Next time you see √2 or π, give them a little respect — they’re proof that numbers can be endlessly fascinating!
See more:
Rational Numbers: NCERT Class 9 Maths
All chapters NCERT Maths Class 9