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is the square root of ten rational or irrational

Guest Jan 10, 2015

Best Answer 

 #1
avatar+26402 
+10

Irrational

.

Alan  Jan 10, 2015
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2+0 Answers

 #1
avatar+26402 
+10
Best Answer

Irrational

.

Alan  Jan 10, 2015
 #2
avatar+91454 
+5

Lets try and prove this.  I am going to do it by contradiction.

 

$$\\Assume \;\;\sqrt{10}\; is\; rational\\\\
then \;\sqrt{10}\;can\;\; be\;\; written\;\; as \;\;\dfrac{p}{q}\\
$where p and q are relatively prime integers (they have no common factors)$\\\\$$

 

$$\\\sqrt{10}=\frac{p}{q}\\\\
10=\frac{p^2}{q^2}\\\\
p^2=10q^2\\\\
So\;\; p^2\;\;$ is a multiple of 10$\\\\
\mbox{so p is a multiple of 10}\\\\
Let\; p=10g\\\\
(10g)^2=10q^2\\\\
100g^2=10q^2\\\\
10g^2=q^2\\\\
So\;\; q^2 $ is a multiple of 10$\\\\
$so q is a multiple of 10$\\\\
$I have discovered that p and q are both multiples of 10$$$

 

Therefore p and q are not relatively prime

therefore the original statement is contradicted 

 

therefore  $${\sqrt{{\mathtt{10}}}}$$    is irrational.

 

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Note:

I have said that since pis a multiple of 10, p must also be a multiple of 10. WHY is so.

Well the prime factors of squared numbers must come in pairs.

Lets look at an example

 

$$\\What\;\; if\;\; p^2=900\\
p^2=30\times 30\\
p^2=3\times 5\times 2 \quad \times \quad 3\times 5\times 2\\
p^2=3\times 3\times 2 \times 2\times 5\times 5\\
$see how the factors have to be in pairs?$\\
p=3\times 2 \times 5\\$$

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10 is the product of 2 prime numbers

Can you see how if  p^2 is a product of 10 then p must also be the product of 10?

Melody  Jan 11, 2015

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