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CCSS.Math:

we're told let a and B be rational numbers and let B be nonzero they have to say let B be nonzero because we're about to divide by B is a over B rational or irrational well let's think about they're both rational numbers so that means that a since it's rational can be expressed as the ratio of two integers so I could write a is equal to M over N and same thing about B I could write B as being equal to P over Q where where m n P and Q are integers are integers by definition of what a rational number is they're telling us these numbers are rational so I can express them as these type of ratios so what is a over B going to be a over B is going to be M over N over P over Q which is equal to M over N if I divide by a fraction is the same thing as multiplying by the reciprocal Q over P let me write that a little bit Q over P which is equal to M Q over N P well M Q is going to be an integer if the product of two integers is going to be an integer and NP is going to be another integer the product of two integers is an integer so I've just said I've just shown that a a over B can be expressed as the ratio of two integers so a over B is for sure in fact I've just proven it to you a over B is for sure going to be rational let's do a few more of these this is interesting all right so now we're saying let a and B be irrational numbers is a over B let a and B be irrational numbers is a over B rational or irrational and like always pause the video and try to think this through and you might want to do some examples of some of the rational numbers and see if you can get when you divide them you can get rational or irrational numbers well let's just imagine a world where let's say that a is equal to I don't know two square roots of two and B is equal to the square root of two well in that world a over B a over would be 2 square roots of 2 over the square root of 2 which would be 2 which is very much a rational number I can express that as a ratio of integers I could write that as 2 over 1 and there's actually an infinite number of ways I can express that as a ratio of two integers so in this case I was able to get a / b to be rational based on a and b being irrational but what if what if instead of what if a was equal to the square root of 2 and b is equal to the square root of let's say b is equal to the square root of 7 well then a / B would be equal to the square root of 2 over the square root of 7 which is still going to be irrational I mean another way to think about it I'm not proving it here but you could think about this is going to be the square root of 2/7 so we have something that's not a perfect square under the radical so we're going to end up with an irrational number so we can we could showed one example where a / b is rational and we showed one example where it is irrational so it can be either way either way let's do a few more of these alright let a be a nonzero rational number is a times square root of eight rational irrational well the key here is if you multiply an irrational number and why is this an irrational number it has a perfect square in it but it's not a perfect square in and of itself square root of eight is square root of eight is equal to the square root of four times two which is equal to the square root of 4 times square root of two which is equal to two square roots of two and this is kind of getting to the punchline of this problem but if I'm if I multiply a rational times an irrational I am going to get an irrational so the square root of eight is an irrational and if I multiply that times a rational number I'm still going to get an irrational number so this is going to be force or irrational let's do one more of these so we're told let a be an irrational number is negative 24 plus a rational or irrational and I won't give I won't give a formal proof here but it'll give you more of an intuitive feel it's nice to just try out some numbers and I encourage you to pause the video and try to think through it yourself well let's just imagine some some values imagine if we a a is irrational a is irrational so what if a was equal to negative pi which is approximately equal to negative 3.14159 and it keeps going on and on forever never never never repeating well then then we would have negative 24 plus a would be equal to negative 24 minus pi which would be approximately negative 27 point one four one five nine the decimal expansion everything to the right of the decimal is going to be the exact same thing as pi so this looks like at least for this example is going to be irrational and let's see if a was square root of two negative 24 plus the square root of two well once again I'm not doing a proof here but intuitively you know this is going to be a this is going to be a decimal it's going to have a decimal expansion that's going to go on forever never repeat and so this would just change that what's to the left of the decimal but not really change what's well it will change what's to the right of the decimal because this is negative but it's still going to go on forever and never repeat and if in fact this was if this was this way then to the right of the decimal you would have the same thing as the square root of two to the left of the decimal you just have a different different value would have what negative negative 25 point whatever whatever whatever and so this is when you add a rational number to an irrational number we prove it in other videos a rational plus an irrational is going to be irrational irrational if you want that proof we have other videos within this tutorial