Definition, Examples | Rational and Irrational Numbers

Numbers are everywhere: you check the time, shop in the store, or do your maths. But did you not know that there are classes of numbers? Rational numbers and irrational numbers are the two most important groups. The knowledge of the distinction between the two helps students develop a better grasp of mathematics.

In this blog, we shall examine the irrational number definition, discuss examples of irrational numbers, study the irrational numbers symbol, and even read through 50 examples of irrational numbers. We shall also contrast the two kinds most clearly in the rational vs irrational numbers.

What Are Rational Numbers?

Any number that can be used as a fraction p/q, where: is a rational numbers.

p and q are integers,

and q ≠ 0.

Rational numbers are:

Whole numbers: 5 (since 5 = 5/1)

Fractions: 3/4, -7/2

Terminating decimals: 0.75 (since 0.75 = 75/100 = 3/4)

Repeating decimals: 0.333… (which is 1/3)

Rational numbers are neat numbers as they can always be expressed in the form of a fraction.

Irrational Number Definition

Now we shall see what an irrational number is.

A number that can not be represented as a fraction p/q, p and q being whole numbers, is known as an irrational number. Their expansion is an endless non-repeating decimal.

Irrational number definition in simple terms:

It is a value which cannot be expressed in the form of a fraction, and the infinite and non-repeating value of its decimal expression.

Irrational Numbers Symbol

The notation commonly employed to denote irrational numbers is Q (or q prime), which is, or so can be expressed, all real numbers not rational.

Rational numbers are written ℚ.

All real numbers, but not ℚ, are referred to as irrational numbers.

So, mathematically:

Irrational numbers = ℝ – ℚ

Examples of Irrational Numbers

The following are some typical examples of irrational numbers:

1. √2 = 1.4142135… (non-terminating, non-repeating)

2. √3 = 1.7320508…

3. π (pi) = 3.141592653… (used in circles)

4. e (Euler's number) = 2.718281828... (exponential functions)

5. Golden ratio (φ) = 1.618033988…

50 Examples of Irrational Numbers

The following is a list to help you have a broader idea. These contain square roots of non-perfect squares, special constants ,and so on:

1. √2

2. √3

3. √5

4. √6

5. √7

6. √8

7. √10

8. √11

9. √12

10. √13

11. √14

12. √15

13. √17

14. √19

15. √20

16. √21

17. √22

18. √23

19. √26

20. √27

21. √29

22. √30

23. √31

24. √33

25. √34

26. √35

27. √37

28. √38

29. √39

30. √41

31. √42

32. √43

33. √46

34. √47

35. √50

36. √51

37. √52

38. √53

39. √55

40. √57

41. √58

42. √59

43. √61

44. √62

45. √65

46. √66

47. √67

48. √68

49. √70

50. π (pi)

Note: The square roots of non-perfect squares are irrational numbers.

Difference Between Rational and Irrational Numbers with Examples

Example of a Rational Number:

0.5 = 1/2

Example of an Irrational Number:

√2 = 1.4142135… (goes on without repeating)

What Is the Use of Irrational Numbers?

Why are such numbers that never-ending? The fact is that irrational numbers can be found everywhere in the world: nature, science, and engineering.

Circles, wheels, and orbits make use of π.

e plays a role in finance, growth, and technology.

Square roots are used in geometry, construction, and design.

Most problems in the real world would not be solvable without irrational numbers.

Fun Facts About Irrational Numbers

The irrational numbers were discovered in ancient Greece. Pythagoreans were appalled to discover that 2 could not be a fraction.

Computers have estimated π to well above 50 trillion decimal places, and it never repeats.

Sunflowers, shells, and even such famous artworks as the Parthenon have the golden ratio ( φ ).

Conclusion

When one is aware of the contrast between rational and irrational numbers, math becomes a lot easier to understand.

Rational numbers are clean and crisp. They can be represented as fractions, and their decimals either cut off or repeat.

Irrational numbers are infinite and unpredictable, and their decimals are infinite and never repeat themselves.

Remembering the irrational number definition, the symbol of the irrational numbers (unofficial ℚ ⁻ 1), and a bit of practice on lists such as the 50 examples of irrational numbers, you will be able to identify them at a glance.

Now, when you next come across 1/4, the square root of 2, or e, you will be able to answer with certainty why they are so interesting members of the irrational number family.

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