i am an combine Professor of mathematics at Juniata college in Huntingdon, PA. I have actually done math research in geometric combinatorics and also linear algebra. I am also very interested in how world learn mathematics and look for avenues to challenge students and also teachers to recognize math much more deeply. *camenga


ns am an combine Professor of math at Houghton university in western new York state. My current research is in linear algebra and also graph theory. I am an analyst by training, therefore I have actually an abiding attention in the genuine numbers and also their properties. I really choose helping human being learn come think favor mathematicians, whatever their backgrounds may be, and also helping teachers aid their students do deep connections.




Hello! My surname is Ivy and I am in seventh grade. I enjoy doing math and drawing and am top top my college MathCounts team. I play piano and also violin. Ns like huge microbes.


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Jonathan is 13 year old and going into eighth grade. That enjoys every music other than pop. And also of course, math is among his favorite subjects. He plays violin in a youth orchestra and also oboe in school band.


Hello! i am Samantha Singh, a climbing seventh grader in ~ Wayzata central Middle School and also second-year college student in the UofM’s Talented Youth mathematics Program. Ns am an recognize of the university of Notre Dame’s NDignite Program, a regime that disclosure leadership amongst high-achieving students. Additionally, I work on the editorial board for KidSpirit magazine and also participate in very first Lego League, receiving nomination for the an international Innovation Award. Sooner or later I hope to design medical tools which will save plenty of human lives.


Abstract

Legend has actually it that the first person in ancient Greece who uncovered that there are numbers the cannot be composed as fractions was thrown overboard indigenous a ship. Centuries later, while we regularly use numbers that cannot be written as fractions, those numbers that can be written as fractions remain an effective tools. What provides fractions so special? We check out how we have the right to recognize the decimal representation of fractions and how fractions deserve to be used to approximate any kind of real number as very closely as us wish.

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On Monday morning, her friend Jordan walks approximately you and says, “I’m thinking of a number between 1 and 100.” gift a great sport, you play along and guess 43. “Nope, as well low!” Jordan declares. “Fine, how about 82?” friend ask. “Too high!” Jordan answers. You keep guessing. 60 is also low. 76 is as well high. 70 is as well low. Feeling pleased the you are gaining closer, friend ask, “How about 75?” “You obtained it!” Jordan replies, and also you march triumphantly off to your very first class of the day.

But ~ class, friend again run into Jordan, who has apparently been thinking around ways come stump you: why rod to confident numbers? What if you additionally allow an adverse numbers? “Now i am reasoning of a number between negative 100 and also 100,” Jordan claims gleefully. You decision to take it the bait, and you quickly uncover that this go not readjust the game much. Girlfriend guess, and by going greater and lower you gain closer and closer to the target. If Jordan’s number is −32, and you have currently figured the end that −33 is too low and also −31 is as well high, then you know the answer is −32. However then girlfriend realize: over there is naught special around −100 and 100! If you begin with a number in between −1000 and 1000, you understand you will ultimately guess the correct number also if that takes a couple of more guesses. Girlfriend march turn off to your 2nd class victoriously, confident that you will certainly be prepared for Jordan’s following challenge.

However, throughout that class, friend realize that you have actually been presume Jordan will always pick an integer. What if fractions space allowed? suppose Jordan picks a number between 0 and 1, for instance 322. You have to guess a number somewhere along the number line from 0 to 1. You try beginning exactly in the middle and guess 12. Jordan speak you your guess is high, so you understand the answer is somewhere on the number line in between 0 and also 12. You guess in the middle again: 14. Jordan claims 14 is still high, so you know the answer have to be ~ above the number line in between 0 and also 14. Continuing with her strategy, you guess 18, 316, 532, 964, …. One representation of this video game is shown in number 1. This seems choose it is acquisition a lengthy time! will certainly you ever guess the right number? probably it would assist if you readjust your strategy. Or space you doomed to be guessing forever?


Figure 1 - A number guessing game.Your friend Jordan asks you to guess a number between 0 and also 1. With each guess, you halve the selection where Jordan’s number can be. The dot at the finish of every line segment is her guess. The position of the number you are trying to guess, 322, is significant by the vertical black color line segment.

A new Strategy: Decimal Expansions

Let united state look at these numbers in a different way and think around them together decimals instead. We have the right to turn a portion into a decimal by dividing the molecule by the denominator. Right here is how it functions for the fraction 716:

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For the an initial step the the division, us ask how plenty of 16’s are in 70. (Really, we room asking how many 1.6’s room in 7.0, but this is equivalent to questioning how many 16’s are in 70). Due to the fact that 16 × 4 = 64, we compose a 4 above the 0 in 7.0. Then we subtract 64 native 70 and get 6 left over. In this case, 6 is referred to as the remainder.

For the following step, we carry down the following 0 native 7.00. Then we ask how many 16’s are in 60. Due to the fact that 16 × 3 = 48, we create a 3 above the second 0. Next, us subtract 48 indigenous 60 to gain a remainder the 12.

We proceed this process, bringing down zeros after every remainder and asking how numerous 16’s space in the result number. ~ we have done this 4 times, we obtain a remainder that 0, which has actually zero 16’s in it. In ~ this point, we space done with our long department and we can say that 716=0.4375. If you room playing the guess-the-number game, you deserve to arrive at this decimal variation of 716 in several short steps. The table below shows a possible way this could happen. In the table, H method your guess: v was too high and also L way your guess was as well low.

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Because the decimal because that the number 716 ends, friend can acquire the specific number by guessing one digit at a time in the decimal. Walk this happen for every fractions? Let united state look in ~ the decimal because that 322.

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Following the same department process, we acquire a 1 on optimal with a remainder that 8, a 3 on optimal with a remainder the 14, a 6 on height with a remainder the 8, a 3 on height with a remainder of 14 … however wait! us have already seen these remainders, and also we understand that the following number on peak is a 6 through a remainder of 14 again. Together we continue to divide, the two repeating remainders the 8 and 14 give us repeating 3′s and also 6′s in the decimal development for 322. This way that if you shot to assumption: v the number 322 one decimal location at a time, you will certainly be guessing forever!

Rational Numbers

All that the numbers us have taken into consideration so far are referred to as rational numbers. A reasonable number is any number that we have the right to write as a fraction ab of two integers (whole number or your negatives), a and also b. This means that 25 is a rational number because 2 and 5 space integers. Also, 3 is a rational number because it deserve to be created as 3=31 and also 4.5 is a rational number due to the fact that it can be composed as 4.5=92. Even if we execute not compose 3 and 4.5 as fractions, they room rational numbers because we deserve to write a portion that is equal to each.

We have seen that part rational numbers, such as 716, have decimal expansions that end. We contact these numbers terminating decimals. Various other rational numbers, such together 322, have decimal expand that keep going forever. However we do understand that also the decimal expand that execute not terminate repeat, for this reason we speak to them repeating decimals.

For any type of rational number ab, the just remainders we can obtain when we compute the decimal are the number 0, 1, 2, 3, …, b − 2, b − 1. Because that example, once we were an altering 322 right into a decimal, the only options we had actually for remainders to be 0, 1, 2, 3, …, 20, 21. Due to the fact that there are only a finite number of remainders, the remainders have to start to repeat eventually. This is true for all fractions whose decimals execute not terminate. Even though there is a repeating sample to the decimals for these fractions, us will never ever guess the specific number in the guessing video game if we room guessing one decimal ar at a time due to the fact that the decimal go on forever. We cannot speak infinitely many digits!

We have the right to go in the reverse direction and readjust decimals to fractions, too! when we have actually a terminating decimal expansion, such as 4.132, us can adjust this to a fraction using location value. The 2 of 4.132 is in the thousandths place, so 4.132=41321000. If us are beginning with a repeating decimal, we have to do a bit more work to find its corresponding fraction. Because that example, take into consideration 0.353535…. Call this number A. The repeating portion 35 has two digits, so we multiply A by 100 to move the decimal over 2 places. This offers 100A = 35.353535…. An alert that every the decimal locations in A and also 100A complement up. We subtract A native 100A to gain 99A. When we subtract the decimals, the 0.353535… is the exact same for both and also is removed in the difference. Therefore, we space left with only entirety numbers!

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We have actually 99A = 35, so as soon as we divide by 99, we obtain A=3599. For any repeating decimal, we deserve to use the same process to find the equivalent fraction. Us multiply by 10, 100, 1000, or everything is essential to relocate the decimal suggest over far sufficient so the the decimal digits line up. Then we subtract and use the an outcome to find the corresponding fraction. This means that every repeating decimal is a rational number!

Irrational Numbers

What if we have a decimal development that does not end, however the digits do not repeat? because that example, look at 0.101001000100001…. In this number, we boost the number of 0s in between each pair of 1s, first having one 0 between, then two 0s, then three 0s, etc. This cannot be a rational number because we recognize the decimals for rational numbers one of two people terminate or repeat. This is an example of an irrational number. One irrational number is any number that we deserve to put on a number line that cannot be composed as a portion of whole numbers. You have actually probably heard about the famed irrational number π = 3.14159…, which gives the ratio of a circle’s circumference come its diameter. While this is a ratio, at the very least one the the circumference or diameter is no an integer, so π is no a reasonable number. Another irrational number is 2=1.41421…, i m sorry is the size of the diagonal of a square whose political parties are size 1.

Going back to our game, all irrational and rational numbers with each other fill increase our number line between 0 and 1. Expect your friend Jordan could pick any type of number between 0 and also 1 and also chose an irrational number because that you to guess. You would certainly likely have a an extremely hard time guessing the number exactly! as with with the repeating decimal development of 322, you can not say infinitely countless digits, therefore this video game seems very unfair.

Let us change the game so you can win! Jordan chooses 3 things: a number because that you to guess, a selection of numbers in which that number lies, and also how close her guess has to be. With these new rules, Jordan choose the number π and also tells friend “I’m thinking of a number in between 2 and also 10. See if you have the right to guess within 0.01 of my number.” In this situation, the game might go choose this:

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In this new version that the game, also if Jordan changes how nearby you need to guess, friend can always eventually get within that distance of π. You just need to get the whole number part and a certain variety of decimal areas correct. Because that example, come be within 0.1 that π, friend only require to acquire the an initial decimal ar correct. To be in ~ 0.01 of π, you need to acquire the an initial two decimal places correct. To be in ~ 0.001 the π, you need to acquire the first three decimal places correct. No matter how close her guess must be, you can win this new game by guessing one decimal location at a time till you have enough decimal places.

As we witnessed earlier, every decimal that terminates is a rational number. If we use this procedure of gaining closer and also closer come an irrational number by guessing more and an ext decimal areas of the number, us can get a rational number the is together close together we favor to our goal irrational number. In our game, this method that no issue what irrational number Jordan chooses nor just how close you need to guess, friend can constantly find a rational number the will fulfill the requirements. In this game, friend can constantly win!

Conclusion

The reason this wake up is the the rational numbers are dense in the real numbers. This way that between any two various real numbers, we can constantly find a rational number. Since real numbers have actually this property, we have the right to approximate any type of irrational number through a rational number. Approximating an irrational number through a rational number is what you are doing in the new game as soon as Jordan choose an irrational number.

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But why would you ever before need to approximate an irrational number with a reasonable number? suppose you are structure a wooden frame for a triangle garden bed in the form of fifty percent the square in number 2. You need to cut a item of hardwood that is 2 feet long. How will you measure the length? due to the fact that 2 is an irrational number, girlfriend cannot usage your tape measure up to measure it exactly! Instead, you will certainly pick a rational number that approximates 2. Friend can select the number of decimal number to include in your development in order to acquire the piece of wood as near in size as you want to 2, the same way you determined your reasonable number come be together close as Jordan wanted you to get in the game.