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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?

• at shouldn't 2/(square root of) 6 be the other way around • 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 • 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.
• How can we use square roots in real life?   