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CCSS.Math:

what number sets does the number three point four zero two eight repeating belong doing before even answering the question let's just think about what this represents and especially what this line on top means so this line on top means the two eight just keep repeating forever so I could express this number as 3.40 to eight for the two eight just keep repeating just keep repeating on and on and on forever I could just keep writing them forever and ever and obviously it's just easier to write this line over the to eight to say that it repeats forever now let's think about what number sets it belongs to well the broadest number said we've dealt with so far is the real numbers and this definitely belongs to the real numbers the real numbers is essentially the entire number line that we're used to using and 3.40 to eight repeating sits someplace over here if this is negative one this is zero one two three four 3.40 to eight is a little bit more than three point four a little bit less than three point four one it would sit right over there so it's definitely sits on the number line it's a real number so it definitely is real it definitely is a real number but the not so obvious question is whether it is a rational number remember a rational number is one that can be expressed as a rational expression or as a fraction if I were to tell you if I were to tell you that P is rational P is rational that means that P can be expressed as the ratio of two integers that means that means that P that means that P can be expressed as the ratio of two integers M over N so the question is can i express this as the ratio of two integers or another way to think of it can i express this as a fraction and to do that let's actually express it as a fraction so let's say that let's define X as being equal to this number so X X is equal to three point four o28 repeating now let's think about what 10,000 X is and the only reason I want 10,000 because I want to move the decimal point all the way to the right over here so 10,000 X 10,000 X what is that going to be equal to well every time you multiply by a power of 10 you shift the decimal one to the right 10,000 is 10 to the fourth power so it's like shifting the decimal over to the right four spaces one two three four so it'll be 34,000 twenty-eight but these two eights just keep repeating so that you'll have you'll still have the two eights go on and on and on and on own after that they just all got shifted to the left of the decimal point by five spaces you could view it that way that makes sense it's a little over it's nearly three and a half if you multiply it by 10,000 you get almost 35,000 so that's 10,000 X now let's also think about 100x and my whole exercise here is I want to get two numbers that when I subtract them and they're in terms of X the repeating part disappears and then we can just treat them as traditional numbers so let's think about what 100x is 100x that moves this decimal point remember the decimal point was here originally it moves it over to the right two spaces so 100x would be 300 let me write it like this it would be 340 point two eight repeating to eight repeating we could have put the two eight repeating here but it wouldn't have made as much sense you always want to write it after the decimal point so we have to write to eight again to show that it's repeating now something interesting is going on these two numbers they're just multiples of X and if I subtract the bottom one from the top one what's going to happen well the repeating part is going to disappear so let's do that let's do that on both sides of this equation let's do it so on the left-hand side of this equation 10,000 X minus 100 X is going to be nine thousand nine hundred X and on the right hand side let's see that the decimal part will cancel out and we just have to figure out what thirty four thousand 28 minus 340 is so let's just figure this out eight is larger than zero so we won't have to do any regrouping there two is less than four so we will have to do some regrouping but we can't borrow yet because we've had a zero over there so the Z and zero is less than three so we have to do some grouping there some borrowing so let's borrow from the four first so if we borrow from the four this becomes a three and then this becomes a ten and then the two can now borrow from the ten this becomes a nine and this becomes a twelve and now we can do the subtraction eight minus 0 is 8 12 minus 4 is 8 9 minus 3 is 6 3 minus nothing is 3 3 minus nothing is 3 so 900 9900 X is equal to 33,000 688 we just subtracted 340 from this up here so we get 33,000 688 now if we want to solve for X we just divide both sides by 9900 we just divide both sides by 9900 divide the left by 9900 divide the right by 9900 and then what are we left with we're left with X is equal to 33,000 688 over 9900 now what's the big deal about this well X was this number X was this number that we started off with this number that just kept on repeating and by doing a little bit of algebraic manipulation and subtracting one multiple of it from another were able to express that same exact X as a fraction now this isn't in simplest terms these probabilities have the I mean they're both definitely divisible by two and it looks like by 4 so you could put this in lowest common form but we don't care about that all we care about is the fact that we were able to represent X we were able to represent this number as a fraction as the ratio of two integers so the number is also rational it is also rational and this technique we did it doesn't it doesn't only apply to this number anytime you have a number that has repeating digits you could do this so in general repeating digits are rational the ones that are irrational are the ones that never ever ever repeat like pi anyway and so the other things I think it's pretty obvious this isn't an integer the integers are the whole numbers that we're dealing with so this is some is in between the integers it's not a natural number or a whole number which depending on the context review two subsets or of integers so Stephenie none of those so it is real and it is rational that's all we can say about it