What are Repeating Decimals? | What Causes Repeating Decimals? | Math with Mr. J

[Music] welcome to math with mr j [Music] in this video i'm going to cover what repeating decimals are and how they happen now repeating decimals are just decimals that repeat without ever ending so they go on forever now this can be one digit that repeats so for example a repeating six or it can be a group of digits that repeats so maybe one two three one two three one two three and those digits repeat without ever ending repeating decimals have a repeating pattern whether it's one digit or multiple digits now we get repeating decimals when we divide and never get a clean cut zero as we go through our division process the pattern just continues on forever and the division never ends repeating decimals can happen when we have a division problem like number one or when we are converting a fraction to a decimal like number two and when we convert a fraction to a decimal we divide so in that case repeating decimals come from division as well let's jump into our examples starting with number one where we have 14 divided by 9 so let's set this problem up 14 divided by 9 and we'll go through our process so fourteen divided by nine how many whole groups of nine out of fourteen well one one times nine is nine subtract we get five now we want to keep this in decimal form so we need to continue on with our division process the goal is to get a clean cut 0 here and that's when our decimal or answer will terminate but in the case of this problem it's going to be a repeating decimal so let's look for a pattern here now since we want this in decimal form we need to place our decimal after the whole number so after the 14 and bring it straight up into our answer once we do that we can extend with placeholder zeros here these do not change the value of the problem this is equivalent to 14 still now we can bring that zero down and we have 50 divided by 9 so how many whole groups of 9 out of 50 well 5 that gets us to 45 multiply 5 times 9 45 subtract we get five so we need to continue on so we can use another zero bring that down and we have 50 divided by nine again so how many whole groups of nine out of fifty well 5 5 times 9 is 45 subtract we get 5 and you're probably seeing a pattern so we can use another zero let me extend this bar here bring that zero down and we have 50 divided by 9 again which is 5. 5 times 9 45 subtract we get 5 we can use another zero and that pattern is going to continue on forever so we get one and then repeating fives that go on forever that pattern will just continue so we can express our answer as one and then five repeating so what we can do we can put this in bar notation is what it's called we put a bar above that five and that shows that the five repeats so our answer one and five repeating let's do another example and move on to number two where we have three elevenths so we're going to take that fraction and convert it to a decimal we do that by using division so we do 3 divided by 11. let's write that out and go through the process so 3 divided by 11 and we start our process by seeing how many whole groups of 11 we can pull out of three well we can't do that so let's put a zero there now in order to continue on and put this in decimal form we need to place our decimal after the whole number and bring it straight up into our answer now we can use a placeholder 0 to keep going and we have 30 divided by 11. so how many whole groups of 11 out of 30 well two 2 times 11 22 subtract we get 8. remember we want to look for a clean cut zero that's when we know that the decimal is going to terminate or end we have an 8 here so we need to continue on let's put another 0 that we can bring down so now we have 80 divided by 11 so how many whole groups of 11 out of 80 well 7. that gets us to 77 let me extend this here 7 times 11 is 77 subtract we get 3 let's use another zero bring that down so we have 30 divided by 11 how many whole groups of 11 out of 30 well two 2 times 11 is 22 subtract we get 8 and you may start to notice a pattern here as we extend with another zero so 80 divided by 11 well that's 7 that gets us to 77 7 times 11 77 subtract we get 3 again we can extend with another zero and that pattern is going to continue on forever so we get 0.27 repeating here and we can express this by writing zero decimal and then the digits that repeat well we have 2 7 2 7 2 7 so the 2 and 7 repeat so we put a bar above the 2 and the 7. if we have a group of digits that are repeating make sure you put the bar above that group that repeats so 0.27 repeating so you can see that as we go through our long division process we have a pattern that continues on forever and that gives us our repeating decimals so hopefully that gives you a better understanding of repeating decimals and how we get repeating decimals so i hope that helped thanks so much for watching until next time peace