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## Algebra (all content)

### Course: Algebra (all content)>Unit 1

Lesson 10: Irrational numbers

# Square roots and real numbers (old)

An old video of Sal where he simplifies square roots in order to determine whether they represent rational or irrational numbers. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• try do a square root of 50 i do not get it
• It is possible to simplify 50 as represented in a square root. Factoring 50, we get:

2 * 5 * 5

All of these are prime numbers. We've factored 50 as far as it will go. You might notice that there are two fives in 50. What happens if we multiply those two fives together? We get 25 as a result of that and 25 so happens to be a perfect square. You can take perfect squares out of the square root symbol.

Since 50 is being represented as a square root to begin with, whatever we take out, we have to take the square root of that because otherwise we change the value of the original expression. In 50, there were two fives that could be multiplied into a perfect square. The reason that those perfect squares are important to begin with is that a perfect square is a number that has a square root that is a whole number. Since we have to take the square root of whatever we take out, it is undoubtedly convenient to be taking out perfect squares so that when we take the square root of them, they don't have decimal numbers.

So, we find the perfect squares in a number, then take the square roots of them and multiply them by whatever we could not take out of the radical symbol. In the case of 50, the only thing that cannot be taken out is that lonely 2. It is not a perfect square, so it would be of little help to try and take that out. We could take out the two fives, which multiply to 25. The square root of 25 is 5, so we can just multiply 5 by the radical symbol that still has the 2.

This gives:

5 * sqrt(2)

Note that sqrt is a way of, in plain text, representing square roots. It is the same thing as putting whatever follows the sqrt in the parentheses, so in this case, it is the same thing as writing a 2 under a radical symbol. I can't write radical symbols since this is plain text and I can't make use of stuff like that, as far as I know.

I hope this helps. :)
• at shouldn't 2/(square root of) 6 be the other way around
• Sal had 2*2*2*3 inside his radical. 2*2=4, 2*3=6. So you have 4*6 inside a radical. The sqrt of 4 is a rational number ( a number that can be expressed as a fraction a/b where b doesn't = 0) so we can go ahead and work with it. We remove the sqrt of four from the radical, leaving us with 2 (the sqrt of 4) times the sqrt of 6.
• what if i need to find the square of a negative number?
• Poemi, you are nearly right. Imaginary isn't irrational.
Both rational and irrational numbers are real, but there are also imaginary numbers that cannot be placed on a number line. i is the square root of -1, and is imaginary. If you wanted to find the square root of -4, that would be the sqrt (2x2x-1), which is 2 sqrt (-1) or 2i.
• Are there negative square roots??
• Two negative numbers multiplied together are positive, so because -4^2 is equal to -4*-4, that is positive. Also, that has the same answer as 4^2, which is 16. So, when you see a question that asks "what is the square root of 25?", there are actually two answers. The first is 5, but the second is -5. By convention, though, you would most likely just write 5 as your answer.
Be careful, though, because a negative number multiplied by a positive number IS negative. So, -4^3(or to any odd power) is negative, because that is the same as -4^2 * -4. As we now know, -4^2 is positive, and of course -4 is negative, so a negative (-4) multiplied by a positive (-4^2) is equal to negative. The number being raised to a power, in this case 4, is called the "base". when squaring (or raising to any even power), whether the base is negative or positive, the answer is ALWAYS positive.
• Does anyone know why 2√6 is irrational? He says "I am not going to explain it in this video," and I was wondering why.
• 2√6 is irrational because √6 is irrational: the product of an irrational number and a rational one (other than 0) is irrational.

So why is √6 irrational? We can prove this by contradiction.

Suppose that √6 is rational. We'll prove that this is impossible, but start by supposing it is. Any rational number can be written as a fraction a / b, with a and b integers. So if √6 is rational, we can say:
√6 = a / b
Even more: it can be written as a fraction in such a way that the fraction is irreducible. So we can assume that a / b is in its simplest form. This in turn means that a and b cannot both be even. At least one of them must be odd, otherwise we can simplify the fraction further until at some point one of them is odd.
Squaring both sides of the equation gives:
6 = a^2 / b^2
Multiply both sides by b^2:
6b^2 = a^2
The left side of this equation is obviously even: 6 is even, so any multiple of 6 is also even. Which means the right side must be even too: a^2 is even. But if the square of a number is even, then that number itself is even too. So a is even.
If a is even, we can write it as a multiple of 2:
a = 2c for some c.
Plug this into the equation above:
6b^2 = a^2 = (2c)^2 = 4c^2
Divide both sides by 2:
3b^2 = 2c^2
The right side is obviously even (a multiple of 2), so the left side must be even as well. 3 is not even, so b^2 must be even. But if the square of a number is even, then that number itself is even too. So b is even.
Hang on.
We just showed that both a and b are even. Which is impossible: we started out by saying that a / b is in its simplest form. If both a and b are even, the fraction is not in its simplest form. We have found a contradiction. Therefore, our original supposition must be wrong, and √6 cannot be rational.

√6 is irrational.

This is a variation of a well-known proof that √2 is irrational.
• What is the square root of 100/225
• The square root of 100/225 is 10/15 or 2/3. We can find this by simplifying the fraction and solving the square roots of the numerator and denominator separately. Here it is:
``100/225 = 4/9   Simplifying``
Now, we solve:
``sqrt(4/9) = sqrt(4) / sqrt(9) = 2 / 3 = 2/3``
• May you please explain square root of 75?

Is it 5 root 3?
• what about PI?, we can express it by 22/7. Is it irrational?
• Pi doe not = 22/7
22/7 is only an estimate for Pi, just like 3.14 is an estimate for Pi.
You can see this if you compare the digits.

Pi is an irrational number -- a non-ending and non-repeating decimal.
Pi = 3.1415926535897932384626...

22/7 is a rational number as it is a ratio of 2 integers. This is the definition of a rational number. In decimal form, 22/7 creates a repeating decimal.
22/7 = 3.142857142857142857...

Hopefully you can see that as soon as you get to thousandths place, these numbers are no longer the same.