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## Algebra 1

### Course: Algebra 1>Unit 15

Lesson 2: Sums and products of rational and irrational numbers

# Worked example: rational vs. irrational expressions

Sal shows how to determine whether the following expressions are rational or irrational: 9 + √(45), √(45)/ (3*√(5)), and 3*√(9). Created by Sal Khan.

## Want to join the conversation?

• How do we know that an integer plus an irrational number yields an irrational number? Is there another video on that?
• We'll do a proof by contradiction. This just means that we show that the false to our statement presents a contradiction.
First, let us assume that an irrational number plus a rational number makes a rational number and make this lead to a contradiction.

If a is rational, b is irrational, and c is rational, we will try to prove that:

``a + b = c``

is rational. If this is true, a = x/y and c = e/f for integers x, y, e, and f. So:

``a + b = c x/y + b = e/f b = e/f - x/y b = ey/(fy) - xf/(fy) b = (ey - xf)/(fy)``

Since the right hand side of the equation is rational, then so is b. But we said that b is irrational! This leads to a contradiction and so the sum must be irrational. Let me know if you need anything clarified.
• Sal cancelled out 3√5/3√5 to get 1. But the order of operations, PEMDAS states that we do the powers before division. So, what happened here? Can anyone please explain me. Thanks!
Sam D
• It doesn't matter because since the numerator and denominator are the same, even if you did use PEMDAS to approximate √5 AND THEN divided them out, you would still get 1. And besides, when we have the square root of a non-perfect square, we leave the answer in radical form (not decimal form), because the decimal form goes on forever like the digits in π. This is because it is irrational. Comment if you have any questions.
• Pi is an irrational number and is the ratio of the circumference (c) over the diameter (d), therefore c/d = Pi. Does this mean that either c or d or both must be irrational or can the quotient of two rational numbers be irrational?
• Do negative square roots exist?
• @CallaJones
By definition, the square root of a negative number does not exist. it instead is called an imaginary number or complex number.

Originally there were only positive integers but over time the concepts of fractions, zero, decimals, negative numbers, irrational numbers, and then certain transcendental numbers (pi or e) were developed to make the number system complete. Leonard Euler invented the idea that we can represent sqrt(-1) with an imaginary number called "i".

For example, the square root of -16 can be expressed as 4i.
sqrt(-16) = sqrt(16) x sqrt(-1) = 4i
• Would you consider Infinity Rational or Irrational?
• Infinity is not a number. It is the concept that there is no largest number. If you think you have found the largest number, you can add 1 and get a still larger number.
Since infinity is not a number, it is not classified as rational or irrational.
• Hmm. Could one multiply irrational numbers to get rational numbers? her it an example.

sqrt(3)*sqrt(3)
sqrt(3*3)
sqrt(9)
3
3/1
• Yes you did it!
(1 vote)
• how do you find square root of 2 ? i know it is irrational but like i want to know the method in detail.
• There is a long division method to find out square roots of numbers up to any decimal place you want. You can search it up.
• Hi guys,

Could anybody help with a question I encountered that I find quite confusing.
"Write as a single fraction . . .
[SQRT(x)] + 1/SQRT(x) . . . . "
Two seperate terms, SQRT(x) and 1/SQRT(x).
I understand many of the rules but I can't get my head around this one.
• find a common denominator. You can multiply anything by 1 will be equal to the same thing. Keep in mind that 1 can be 432tvx^2/432tvx^2
or anything where the topand bottom are equal
• What about the question (√2 - √3) ^ 2 ?

Write in Racial Form?

I know that it is not very relevant but there is nowhere else to ask it!

(1 vote)
• Radical form means you use the radical symbol where needed rather than exponential form.
-- Exponential form: 2^(1/2)

How to do: (√2 - √3)^2
Did you use the hints?
You need to multiply 2 binomials, which means you use FOIL. (a-b)^2 = (a-b)(a-b) = a^2-ab-ab+b^2 = a^2-2ab+b^2

Remember to simplify the radicals. For example: √2√2 = √4 = 2

Give it a try. Comment back with questions.