Hello, I want to teach a little bit about irrational numbers today
The better to understand what irrational number is I think first is best To talk a little bit about what a rational number is
An irrational number is any number that can be written as a fraction or repeating decimals For example 3 over 4 would be a rational number and you could write 3 over 4 as a decimal 0.75
5 is a number it can be written as fraction 5 over 1 so therefore to a rational number:
1/2 is another fraction and I used the example, this is was 0.33 repeated
If it's a repeating decimal it can be written as a fraction this one is the fraction of 1 over 3 alright as long as it's a repeating Pattern
What irrational numbers are are any number that cannot be written as a fraction or repeating decimal
For example the square root of 2 the square root of 15, the square of 7 these are all members that are repeated, but not in the pattern pi is another one 3.14, 3.15 you know continues on but it doesn't really have a pattern right, so it can't be written in a Repeating decimal or in a fraction basically if you can't read it in a fraction it isn't irrational numbers here just some examples
Now that we know what are irrational number is our job is to find if 72 is rational or irrational number to do that the first step is for us to actually factor the square root of 72
So I'll take 72 and we will find factors of 72 they are perfect squares.
If you remember from perfect squares. Perfect squares are numbers that can be divided evenly by the same number like for example 16 divided by 4 is equal to 4 so 4 times 4 is 16 are rational square.
So we're gonna do is look for some perfect squares of 72. 72 is divisible by 4, 72 divided by 4 is 18 but there are some other larger perfect Squares that we can divide 72 by 72 divided by 16 is four-and-a-half so that's not going to work for us
Lets try 72 divided by 9 that works out for us 72 divided by 9 will go ahead and do that So that means 72 divided by 9 is 8 so that means The square root of 9 times square root of 8
Now we see here that we have 8, Now 8 can be divided further because it is divisible by 4 and 4 is a perfect square
So we have 2 times 4 is 8, 8 times 9 is 72 so what we have here is or factoring down The 72 because we have 2 it means that we could have divided 72 in the first place 72 divided by 36 but it′s all right that we didn't know That We found it this way by factoring Throughout.
Square root of 9 Is 3 the square root of 4 is 2, we have Square root of 2 to last
3 times 2 is 6
Now the question is, is this rational or irrational, well and how the square of 2 here at the end so that means that we can't write it as a fraction and so this is an irrational number
We found out that this square of 72 is irrational
Now we go to the next, is the square root of 324 rational or irrational
We're gonna look for some perfect squares that are factors of 324 let me start something big like 81 we find that 81 is a factor of 324, 81 times 4
Now this tells me something the square root of 81 is 9 the square root of 4 is 2, 9 times 2 is 18 so what that means is that 18 times 18 is actually 324
So fracturing about like this has been able to get me the solution there and we solved and found out that this is irrational number square root of 324 is 18
We're gonna look at square root of 567
to find the square root of this number we are going to search for factors that are perfect squares so we're gonna start out with something pretty big 567 it's a pretty big number so we'll start out like 49 it is a perfect square divide 567 by 49 we find out that is not a factor of 567
Let's try another one that have a big One, will be 64
And again 567 is not divisible by 64 so this will try moving down a little bit.
Divided by 36 together again is not a factor of 36
So let to 16, 567 is not divisible by 16.
567 divided by 4 nope. 567 divided by 9 all their we go 567 divided by 9 is 63 times 9
And if you look at 63, 63 is also evenly divisible by 9 so we can divide 63 by 9 and write that one out as well, so have the square root of 9 times square root of 7 times square root of 9
We factor those who square of 9 is 3 Square of 7 and square of 9 is 3 again
So 3 times 3 is 9 and square of 7
We discovered 9 and the square of 9 so 567 is divisible by 81 as well, but I have it memorized the factors at 567 so I was just practicing divided different things by 567 ok, just happened, but I didn't do the right ones that's okay because we're trying to find if numbers are rational or irrational it's gonna take a little bit of guesswork and it's gonna take a little bit as finding what works for us and as soon as we find one factor like we found 9 was a factor here Tends to help because it'll break down these number even further to the point where we can see some factors in these numbers cuz I know the factors of 63 I don't have to know all the factors of 567 alright
So here's what I've done and I factories down the final answers is 9 times Square of 7 the square of 7 cannot be written in a fraction so therefore it is an irrational number alright
And that is basically the way that you find out if the numbers, rational or irrational will take that same that number try and find some perfect square factors up that number and factored down as much as you can
I hope that this lesson is that helpful for you on finding if the numbers are rational or irrational