# Rationalise Denominator

• Sep 11th 2010, 12:17 PM
webguy
Rationalise Denominator
Hi,

In order to rationalise the fraction $\frac{1}{1+\sqrt{3}}$ I must multiply the numerator and denominator by $1-\sqrt{3}$. I do not understand why this is so. Why does the sign only change for the surd and not the rational number (1) which is added to it?

Thanks for any help (Happy)
• Sep 11th 2010, 12:20 PM
undefined
Quote:

Originally Posted by webguy
Hi,

In order to rationalise the fraction $\frac{1}{1+\sqrt{3}}$ I must multiply the numerator and denominator by $1-\sqrt{3}$. I do not understand why this is so. Why does the sign only change for the surd and not the rational number (1) which is added to it?

Thanks for any help (Happy)

(a+b)(a-b) = a^2 - b^2
• Sep 11th 2010, 12:21 PM
Moo
Hello,

Recall the identity $(a+b)(a-b)=a^2-b^2$.

In order to use this identity, you only need one among a or b to change its sign.

It's not a formal explanation but I hope you will understand :D
• Sep 11th 2010, 12:25 PM
webguy
Would anyone mind going into deeper detail? I would like to understand this logic in its completeness.
• Sep 11th 2010, 12:37 PM
undefined
Quote:

Originally Posted by webguy
Would anyone mind going into deeper detail? I would like to understand this logic in its completeness.

I'm not sure what you're asking. You mean this?

$(a+b)(a-b) = a^2 -ab + ab -b^2 = a^2-b^2$
• Sep 11th 2010, 12:46 PM
webguy
So why does that identity help us rationalise that fraction?
• Sep 11th 2010, 12:51 PM
Wilmer
Code:

a + b a - b ===== a^2 + ab     - ab - b^2 ============== a^2      - b^2
Just a multiplication:
a times (a + b) = a^2 + ab
-b times (a + b) = -ab - b^2
• Sep 11th 2010, 12:58 PM
undefined
Quote:

Originally Posted by webguy
So why does that identity help us rationalise that fraction?

For integers a and b with b non-negative, if you have denominator $a + \sqrt{b}$ and you multiply it by $a - \sqrt{b}$ you get $a^2 - b$ which is an integer. Thus we have made the denominator rational.
• Sep 11th 2010, 02:35 PM
webguy
Thank you! Now it makes sense!