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### Course: Algebra 1>Unit 15

Lesson 3: Proofs concerning irrational numbers

# Proof: there's an irrational number between any two rational numbers

Sal proves that when given any two rational numbers, no matter how close, we can find an irrational number that lies between them. Created by Sal Khan.

## Want to join the conversation?

• why 1/sqrt(2) = sqrt(2)/2 ?
• I found another way, which uses basic lessons from Khan Academy about roots and fractions.

1/sqrt(2) = 1/sqrt(4/2) = 1/(2/sqrt(2)) = 1 * sqrt(2)/2 = sqrt(2)/2
• May I add a simple logic to all of this? This way more people will understand. Can you confirm that my logic holds true, Sal?
Here is one way of looking at the claim we are attempting to prove
Question: Can we cover each distance on the number line with the set of rational numbers, or is the set of rational numbers insufficient?
It might originally appear that the set of rational numbers suffices, for, after all, we can always make smaller and smaller rationals. Say we start at 1 unit distance, and we divide this distance in 2 repeatedly, we should get smaller and smaller units, until it becomes infinitely small.
However, by finding that in 1 place on the number line, there exists a irrational between 2 rationals, we know that the set of rationals is insufficient to cover all distance on the number line.
Because there is nothing special about one specific place in the number line over any other place (accomplished technically by adding r1 and multiplying by r2-r1), we know that between all 2 rationals there exists an irrational.
• You've given a reasonable summary of Sal's proof.
• How to prove rational number+irrational number is an irrational number
• First prove a rational - rational = rational. A rational is a fraction a/b where a and b are natural numbers. Let a/b and c/d be two rational numbers...
a/b + c/d = (ad + bc)/bd
ad + bc = natural number
bd = natural number
so (ad + bc)/bd is a rational number
So a rational - a rational = rational...REMEMBER THIS
Now let a/b be a rational number and x an irrational number
Suppose a/b + x = c/d where c/d is a rational number
So we have...
a/b + x = c/d
x = c/d - a/b
But you know (from our first proof) that c/d - a/b is a rational number.
So, x is a rational number AND x is an irrational number. Contradiction!

Therefore, our assumption that a rational + an irrational = rational is false.
Therefore, a rational + an irrational = irrational

This method is called reductio ad absurdum or proof by contradiction. Read it up, it's very interesting.
• Why do I need to prove this? What if I just write a number like 84,5426578422158456622 Is it not an irrational number?
• ``Good question !First of all, since your number can be written with a finite amount of rational digits it is definitely not irrational. Irrational numbers can not be written with a finite amount of non repeating digits or an infinite amount of repeating digits, i.e. they do not show a pattern when expressed with rational numbersThen to the second point, "Why": Saying things like "What if ..." or "is it not..." is not enough for a mathematical proof. A proof in Math has to be absolutely stable against all reasonable doubt. This comes from a peculiarity of mathematical proofs: They are disproven as soon as you find ONE counterexample. Unlike e.g. in Medicine, where basically all "security" comes from statistics, a mathematical proof is either perfect (=absolutely secure against all counter examples) or not valid at all.``
• proof important for sat or not?
• The SAT is mainly short answers, so proofs aren't really needed, usually just the result
• so if the square root of like 49 is 7 then is it irrational or rational
• It's rational because after you take the square root of 49 and get 7, the 7 could be written as a ratio, or fraction, equal to 7/1.
• Is there an irrational number between 0.999999999... and 1? I have thought about this problem for a while but can't find an irrational number between them although Sal says there is. Please help.
• It's because by simple arithmetic logic, 0.99999..... is equal to 1. Let's see how:
``let x = 0.9999....10x = 9.9999...10x = 9 + 0.9999...10x = 9 + x9x = 9x = 1``

Now whether it's really true or not, beyond my scope. It means that 0.9999... is so, so close to 1, that it equals 1. So there wouldn't be an irrational number between them, as both of them are the same number.

Hope it helps. :)
• This one is much easier to understand than the last proof for dumb like me :)
• Can I find a countably infinite number of irrationals between any two rational numbers ?