give 30 examples of irrational numbers, both fractions and decimals
Here are 30 examples of irrational numbers, both in fractional and decimal forms:
1. π (pi) = 3.14159265358979...
2. √2 (square root of 2) ≈ 1.41421356237310...
3. √3 (square root of 3) ≈ 1.73205080756887...
4. √5 (square root of 5) ≈ 2.23606797749979...
5. √7 (square root of 7) ≈ 2.64575131106459...
6. √10 (square root of 10) ≈ 3.16227766016838...
7. √11 (square root of 11) ≈ 3.31662479035540...
8. √13 (square root of 13) ≈ 3.60555127546399...
9. ∛(2) (cube root of 2) ≈ 1.25992104989487...
10. ∛(3) (cube root of 3) ≈ 1.44224957030741...
11. ∛(5) (cube root of 5) ≈ 1.70997594667670...
12. ∛(7) (cube root of 7) ≈ 1.91293118277239...
13. ∛(10) (cube root of 10) ≈ 2.15443469003188...
14. ∛(11) (cube root of 11) ≈ 2.22398009056931...
15. ∛(13) (cube root of 13) ≈ 2.35133468772076...
16. φ (Golden ratio) ≈ 1.61803398874989...
17. e (Euler's number) ≈ 2.71828182845904...
18. ln(2) (natural logarithm of 2) ≈ 0.69314718055995...
19. ln(3) (natural logarithm of 3) ≈ 1.09861228866811...
20. ln(5) (natural logarithm of 5) ≈ 1.60943791243410...
21. ln(7) (natural logarithm of 7) ≈ 1.94591014905531...
22. ln(10) (natural logarithm of 10) ≈ 2.30258509299404...
23. ln(11) (natural logarithm of 11) ≈ 2.39789527279837...
24. ln(13) (natural logarithm of 13) ≈ 2.56494935746153...
25. 1/π ≈ 0.31830988618379...
26. 1/√2 ≈ 0.70710678118655...
27. 1/√3 ≈ 0.57735026918963...
28. 1/√5 ≈ 0.44721359549996...
29. 1/√7 ≈ 0.37796447300923...
30. 1/√10 ≈ 0.31622776601684...
Note: The decimal approximations provided are rounded off. The examples listed are generally accepted irrational numbers; however, the exact values of some "named" irrational numbers, such as π and √2, are unknown and are continuously calculated with increasingly accurate approximations.
easiest way to tell the difference between irrational and rational numbers
The easiest way to tell the difference between irrational and rational numbers is to look at their decimal representation:
1. Rational numbers: Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. When represented as decimals, rational numbers are either terminating (finite) decimals or repeating decimals.
Examples: 1/2 = 0.5, 1/3 = 0.3333..., 0.25 = 1/4, 2/5 = 0.4.
2. Irrational numbers: Irrational numbers cannot be expressed as a fraction and do not terminate or repeat when represented as decimals.
Examples: √2 = 1.41421356..., π (pi) = 3.14159265..., e = 2.71828182..., √3 = 1.73205080...
So, if a number has a decimal representation that terminates or repeats, it is rational. If the decimal representation goes on forever without repeating, it is irrational.
is this a rational or irational decimal: 2.38483948090....
To determine if the decimal 2.38483948090... is rational or irrational, we need to check if it terminates or repeats. Unfortunately, you have not provided enough digits to determine this.
If the decimal eventually terminates or begins to repeat at some point, it would be a rational number. However, if the decimal goes on indefinitely without any repetition or termination, it would be an irrational number.
Without additional information, it is not possible to definitively classify 2.38483948090... as either rational or irrational.