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# Intro to rational & irrational numbers

CCSS.Math:

## Video transcript

so let's talk a little bit about rational rational numbers rational numbers and the simple way to think about it is any number that can be represented as the ratio as the ratio of two integers is a rational number so for example any integer is a rational number one can be represented as 1 over 1 or as negative 2 over negative 2 or as 10,000 over 10,000 in all of these cases these are all different representations of the number 1 ratio of two integers and I she obviously can have an infinite number of representations of 1 in this way the same number over the same number the number negative 7 could be represented as negative 7 over 1 or 7 over negative 1 or negative 14 over positive 2 and I can go on and on and on and on so negative 7 is definitely a rational number can be represented as the ratio of two integers but what about things that are not integers for example let us imagine oh I don't know let's imagine 3 3 point 7 5 how can we represent that as the ratio of two integers well 3.75 you could you could rewrite that as 375 over over 100 which is the same thing as 750 over 200 or you could say 3 point 7 5 is the same thing as 3 and 3/4 as 3 and 3/4 so let me write here 3 and 3/4 which is the same thing as that's 15 over 4 4 times 3 is 12 plus 3 is 15 so you could write this this is the same thing as 15 over 4 or we could write this as negative 30 over negative 8 I just multiply the numerator later here by negative two but just to be clear this is clearly rational I'm giving you multiple examples of how this can be represented as the ratio as the ratio of two integers now what about repeating decimals well let's take maybe the most famous of the repeating decimals let's say you have zero point three three three just keeps going on and on forever which we can denote by putting that little bar on top of the three this is zero point three repeating and we've seen and later we'll show that how you can convert any repeating decimal into a rational as the ratio of two integers this is clearly one over three or maybe you've seen things like zero point six repeating which is two over three and there's many many many other examples of this and we'll see any repeating decimal not just one digit repeating as many even if it has a million digits repeating as long as the pattern starts to repeat itself over and over and over again you can always represent that as the ratio as the ratio of two integers so I know what you're probably thinking hey Sal you you've just included a lot you've included all of the integers you've included all of non repeating decimals or you all you've included all of finite and non repeating decimals and you've also included repeating decimals what is left are there any numbers that are not rational and you're probably guessing that there are otherwise people wouldn't have taken the trouble of trying to label these as rational and it turns out as you could imagine that actually some of the most famous numbers in all of mathematics are not rational and we call these numbers irrational irrational irrational numbers irrational numbers irrational numbers and I've listed just a few of the most noteworthy example pi the ratio of the circumference of the the ratio of the circumference of the diameter of a circle is an irrational number it never terminates it goes on and on forever and it never repeats e same thing never terminates never repeats it comes out of continuously compounding interest it comes out of complex analysis shows up all over the place square root of two irrational number five the golden ratio irrational number so these things that really just pop out of nature many of these numbers are irrational now you might say okay are these irrational these are just kind of these special these special kind of numbers but maybe most numbers are rational and you know Sal's just picked out some special cases here but the important thing to realize is they do seem exotic and they are exotic in certain ways but they aren't uncommon it actually turns out that there is there's always an irrational number between any two rational numbers and in that well we could go on and on there's actually an infinite number but it but there's at least one it so that gives you an idea that it's you can't really say that there are fewer irrational numbers than rational numbers in a future video we'll prove that you give me you give me two rational numbers rational one rational two rational two there's going to be at least one irrational number between those which is a kind of a neat result because irrational numbers seem to be kind of exotic another way to think when I took square root of 2 but you take the square root of any non perfect square you're going to end up with an irrational number you take the sum of an irrational and a rational number and we'll see this later on we'll prove it to ourselves the sum of an irrational and a rational is going to be irrational the product of an irrational and irrational is going to be irrational so there's a lot a lot a lot of irrational numbers out there