# Why are these two values the same?

Why do we change tan 30 = 1/sqrt3 to sqrt3/3 ?

Why can't I leave tan 30 as 1/sqrt3

This is for a right angled triangle with a hypotenuse of 2.

Thanks (in advance) for the help.

### 9 Answers

- billrussell42Lv 75 months agoBest answer
It's just a convention, to not leave a radical in the denominator.

1/√3 is correct, as is √3/3

as to why they are the same, the problem you posted explains that very well.

- Anonymous5 months ago
because you are multiplying both the numerator and the denominator by the same numerical value

- Φ² = Φ+1Lv 75 months ago
This has a name: "Rationalizing the Denominator" and you can find plenty of sites offering methods and reasons.

As to why 1/√3 is the same as √3/3, it is because the right triangle with sides 1, √3, and 2 is proportional / similar to the right triangle with sides √3, 3, and 2√3.

Long story short, a result is not considered "simplified" with an irrational denominator, but the values are the same.

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- llafferLv 75 months ago
When you simplify irrational numbers we always want the denominator to be a rational number, moving the irrational portion to the numerator.

1/√3

If we multiply both halves by √3 we don't change the value of the expression, but move the irrational component to the numerator:

√3 / √9 = √3 / 3

- Anonymous5 months ago
It's to make things more difficult for students who are struggling with math.

- D gLv 75 months ago
It is to simplify math later..

when you have a squareroot .. in the bottom of a fraction .. its just general rule to multiply by itself or itself which is 1 and gives you a more easy value to work with with an integer on the bottom..

- DixonLv 75 months ago
As billrussel42 says, it is just convention to "simplify" by putting radials (roots) in the numerator. I'm not sure of the exact reasoning behind this but it does mean that you don't get caught out when collecting terms and comparing expressions, eg

1/√3 + 5 + 2√3

= 5 + √3/3 + 2√3

[thinking of √3/3 as (1/3)√3]

= 5 + (7/3)√3

and we should all get the same answer apart from maybe the order of the terms.

- AmyLv 75 months ago
> Why are the values the same

The image you uploaded tries to answer that.

√3/√3 = 1

We can multiply any number by 1 without changing its value. Therefore we can multiply by √3/√3.

To multiply fractions, you multiply the numerators and multiply the denominators.

> Why do we change it?

This is just an agreed-upon convention in how we write numbers. Like writing the terms of a polynomial with the highest exponent first.

Mathematically, it is equally correct to write 1/√3 or √3/3 or √(1/3), just as it is equally correct to write x + x^3 instead of x^3 + x.

Assuming that you're planning to do anything with the result of your calculation, or even just use it in some more calculations, it's often easier to deal with a fraction that has a rational denominator. √3 is approximately 1.732. How easily can you measure something to a length of 1/1.732 meters? How about 1.732/3 meters?

Just use 57.735cm in both cases!

you can leave a radical in the denominator, it's just a convention not to.

as I said, 1/√3 is a correct answer.