Is Every Root of 2 Higher Than 1 Irrational?

In summary, The conversation discusses the irrationality of different roots, such as the square root of 2, and how the same idea can be applied to other roots above 2, such as the cube root or quartic root. It is confirmed that this is correct, and methods such as prime factorization and the rational root theorem can be used to prove the irrationality of other roots that are not natural numbers. It is also noted that the only exception is the "1" root of 2, which is rational.
  • #1
Char. Limit
Gold Member
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I was thinking on the square root of 2 being irrational proof... and I got the idea that you could use the same idea for every root higher than two. The cube root, the quartic root, the quintic root, etcetera. (Obviously assuming the roots are natural numbers.)

As a reassurance I'm not crazy, is this correct?
 
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  • #2
You are not crazy. At least for this particular reason. :wink:
 
  • #3
Good to know.

Is there a way to prove that other roots that aren't natural numbers are irrational?
 
  • #4
Prime factorization.
 
  • #5
Also, rational root* theorem.

*: as in, root of a polynomial
 
  • #6
So... are all roots of two that aren't 1 irrational?
 
  • #7
Almost. Of course the "1" root of 2, 21/1= 2 is rational! If n is a positive integer, greater than 1, then 21/n is irrational.
 

FAQ: Is Every Root of 2 Higher Than 1 Irrational?

1. What does it mean for a number to be irrational?

Irrational numbers are numbers that cannot be expressed as a ratio of two integers. This means they cannot be written as a fraction with a finite number of digits in the numerator and denominator. They are non-repeating and non-terminating decimals.

2. Why is it significant that every root of 2 is irrational?

This statement is significant because it proves that there are numbers that cannot be expressed as a ratio of two integers. It also shows that there are numbers that cannot be represented by a decimal with a finite number of digits. This has many implications in mathematics and has been studied extensively by mathematicians.

3. How was it proven that every root of 2 is irrational?

The proof for this statement was first given by the ancient Greek mathematician, Pythagoras. He showed that the square root of 2 cannot be expressed as a ratio of two integers by using a geometric proof. This proof has been refined and expanded upon over the years by other mathematicians.

4. Are there any other numbers besides the root of 2 that are irrational?

Yes, there are infinitely many irrational numbers. Some examples include pi, e, and the square root of any non-perfect square. In fact, the majority of numbers are irrational, and it is only a small subset that can be expressed as a ratio of two integers.

5. How is the concept of irrational numbers used in real life?

Irrational numbers are used in many areas of mathematics and science, including geometry, physics, and engineering. In real life, they are used to represent values that are not whole numbers, such as the circumference of a circle or the growth rate of a population. They are also important in computer science and cryptography.

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