Is \sqrt{2} + \sqrt{3} + \sqrt{5} irrational?

In summary, the conversation discusses whether the expression \sqrt{2} + \sqrt{3} + \sqrt{5} is rational. The conversation uses the fact that the sum of an irrational number and a rational number is irrational, and applies it to the given expression. The conversation also presents a proof by contradiction to show that the expression is irrational.
  • #1
Mantaray
17
0

Homework Statement



Is [tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] + [tex]\sqrt{5}[/tex] rational?

Homework Equations



If n is an integer and not a square, then [tex]\sqrt{n}[/tex] is irrational

For a rational number a and an irrational number b,

a + b is irrational
a * b is irrational if a is not equal to 0

The Attempt at a Solution



Assume that [tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] + [tex]\sqrt{5}[/tex] = x, with x being a rational number.

[tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] = x - [tex]\sqrt{5}[/tex]
=> ([tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex])2 = (x - [tex]\sqrt{5}[/tex])2
=> 2 + 2[tex]\sqrt{6}[/tex] + 3 = x2 - 2x[tex]\sqrt{5}[/tex] + 5
=> 2[tex]\sqrt{6}[/tex] = x2 - 2x*[tex]\sqrt{5}[/tex]
=> (2[tex]\sqrt{6}[/tex])2 = (x2 - 2x[tex]\sqrt{5}[/tex])2
=> 24 = x4 - 4x3*[tex]\sqrt{5}[/tex] + 20x2

- 4x3*[tex]\sqrt{5}[/tex] is irrational because 4x3 is rational.
x4 - 4x3*[tex]\sqrt{5}[/tex] + 20x2 is thus irrational.

The left hand side of the equation is rational, as 24 is a rational number.

This is a contradiction, thus our assumption was false, x cannot be a rational number.

[tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] + [tex]\sqrt{5}[/tex] is thus irrational

Is this a valid proof, or should the equation be worked out further?
 
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  • #2
Skimming it, the proof looks good.
 
  • #3
I looked at some threads on the internet, but worked the equation out to an eighth degree equation, so I wasn't sure this way was actually correct. Thanks a lot!
 

1. What is an irrational number?

An irrational number is a type of real number that cannot be expressed as a ratio of two integers. They are numbers that go on forever without repeating a pattern, and their decimal representation never ends or settles into a repeating pattern.

2. How are irrational numbers different from rational numbers?

Irrational numbers are different from rational numbers in that they cannot be expressed as a fraction of two integers. Rational numbers, on the other hand, can be expressed as a ratio of two integers.

3. Can irrational numbers be negative?

Yes, irrational numbers can be negative. Examples of negative irrational numbers are -π and -√2.

4. What are some examples of irrational numbers?

Some examples of irrational numbers include π, √2, √3, √5, and e. These numbers cannot be expressed as a ratio of two integers and their decimal representation goes on forever without repeating a pattern.

5. How are irrational numbers used in science?

Irrational numbers are used in various scientific fields, such as physics, engineering, and mathematics. They are essential for accurately representing physical quantities and calculating precise measurements. For example, the value of π is used in many equations and formulas in physics and engineering.

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