Proove that the cubic root of 2 + the square root of 2 is irrational

In summary, the cubic root of two plus the square root of two is irrational because the sum of two irrational numbers with infinitely many decimals will always be irrational. This can be proven by showing that the sum of a rational and an irrational number will always be irrational.
  • #1
ehj
79
0
How do you show that the cubic root of two + the square root of two is irrational? I can easily show that each of these numbers is irrational, but not the sum :/.
 
Physics news on Phys.org
  • #2
ehj said:
How do you show that the cubic root of two + the square root of two is irrational? I can easily show that each of these numbers is irrational, but not the sum :/.

If both of the numbers have infitely many decimals, there can never be a terminating digit for their sum.
 
  • #3
Thats not a proof? And besides, doesn't 1 - sqrt(2) and 1 + sqrt(2) both have infinately many decimals, nevertheless their sum is rational, 2.
 
  • #4
asleight said:
If both of the numbers have infitely many decimals, there can never be a terminating digit for their sum.

But 1/3 = .333333... has infinitely many decimal places and it is rational as are 1/2 = .50000... (can also be written as .49999999...) and 1/5 = .200000 (can also be written as .1999999...).
 
  • #5
I figured it out. Posting solution in case sombody might run into the same problem in the future :P

I assume 2^(1/3) + 2^(1/2) = a , where a is rational

=> 2=(a-2^(1/2))^3 <=> 2 = (a^3 + 6a) + sqrt(2)(-3a^2 -2)

Which is a contradiction since sqrt(2)(-3a^2 -2) is an irrational multiplied by a non-zero rational, which can be proved to always be irrational, and the sum of a rational (a^3 + 6a) and an irrational can be proved to always be irrational, and above cannot equal 2 since 2 is rational.
 
Last edited by a moderator:

What does it mean for a number to be irrational?

An irrational number is a number that cannot be expressed as a ratio of two integers. In other words, it cannot be written as a fraction in the form of a/b, where a and b are integers. Irrational numbers are non-repeating and non-terminating, meaning their decimal representation goes on forever without a repeating pattern.

What is the cubic root of 2 + the square root of 2?

The cubic root of 2 + the square root of 2 is a mathematical expression that represents the sum of the cubic root of 2 and the square root of 2. This can also be written as ∛2 + √2.

Why is proving the cubic root of 2 + the square root of 2 irrational important?

Proving that the cubic root of 2 + the square root of 2 is irrational is important because it provides evidence for the existence of numbers that cannot be expressed as a ratio of integers. This has significant implications in mathematics and other fields, such as computer science and cryptography.

How do you prove that the cubic root of 2 + the square root of 2 is irrational?

To prove that the cubic root of 2 + the square root of 2 is irrational, you can use a proof by contradiction. Assume that the expression is rational, meaning it can be written as a fraction. Then, by squaring both sides and simplifying, you will reach a contradiction. This shows that the original assumption was false and the expression is indeed irrational.

What are the real-world applications of understanding irrational numbers?

Understanding irrational numbers has many real-world applications, such as in physics, engineering, and finance. In physics, irrational numbers are used to describe natural phenomena, such as the golden ratio in the structure of plants. In engineering, they are used in design and construction, such as in the angles of bridges and buildings. In finance, irrational numbers are used in the calculation of interest rates and stock market fluctuations.

Similar threads

Replies
31
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
659
  • Calculus and Beyond Homework Help
Replies
3
Views
805
  • Calculus and Beyond Homework Help
Replies
3
Views
776
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
222
  • Calculus and Beyond Homework Help
Replies
3
Views
9K
Back
Top