ns am an associate Professor of math at Juniata college in Huntingdon, PA. I have actually done math research in geometric combinatorics and linear algebra. I am also really interested in how human being learn mathematics and also look for methods to challenge students and also teachers to understand math more deeply. *camenga

i am an associate Professor of mathematics at Houghton college in western brand-new York state. My recent research is in linear algebra and graph theory. I am an analyst through training, so I have actually an abiding attention in the genuine numbers and also their properties. Ns really prefer helping world learn to think like mathematicians, every little thing their backgrounds might be, and also helping teachers assist their students make deep connections.

Hello! My surname is Ivy and also I to be in saturday grade. I enjoy doing math and drawing and am on my college MathCounts team. Ns play piano and also violin. Ns like gigantic microbes.


Jonathan is 13 year old and also going right into eighth grade. That enjoys all music other than pop. And of course, math is among his favourite subjects. That plays violin in a youth orchestra and oboe in college 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. I am an admit of the university of Notre Dame’s NDignite Program, a program that promotes leadership among high-achieving students. Additionally, I occupational on the editorial plank for KidSpirit magazine and participate in first Lego League, receiving nomination for the an international Innovation Award. At some point I hope to design medical gadgets which will certainly save plenty of human lives.


Legend has actually it that the an initial person in old Greece who found that there are numbers the cannot be composed as fractions to be thrown overboard from a ship. Centuries later, if we regularly use numbers that cannot be created as fractions, those numbers that can be composed as fountain remain powerful tools. What provides fractions therefore special? We explore how we deserve to recognize the decimal representation of fractions and also how fractions have the right to be used to approximate any type 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 reasoning of a number in between 1 and 100.” being a an excellent sport, girlfriend play along and guess 43. “Nope, also low!” Jordan declares. “Fine, how about 82?” friend ask. “Too high!” Jordan answers. You save guessing. 60 is too low. 76 is also high. 70 is too low. Feeling pleased that you are gaining closer, girlfriend ask, “How about 75?” “You got it!” Jordan replies, and also you in march triumphantly turn off to your very first class of the day.

But after ~ class, girlfriend again run into Jordan, who has actually apparently been thinking about ways to stump you: why stick to optimistic numbers? What if you likewise allow an unfavorable numbers? “Now ns am thinking of a number between an unfavorable 100 and 100,” Jordan claims gleefully. You decision to take it the bait, and also you quickly discover that this go not change the game much. Girlfriend guess, and also by going higher and lower you get closer and closer to the target. If Jordan’s number is −32, and also you have already figured out that −33 is as well low and also −31 is as well high, climate you recognize the price is −32. Yet then girlfriend realize: over there is nothing special around −100 and also 100! If you start with a number between −1000 and also 1000, you know you will ultimately guess the correct number even if it takes a few more guesses. Girlfriend march turn off to your 2nd class victoriously, confident the you will certainly be prepared for Jordan’s next challenge.

However, throughout that class, girlfriend realize the you have been suspect Jordan will always pick an integer. What if fractions are allowed? suppose Jordan choose a number between 0 and also 1, for example 322. You have to guess a number somewhere along the number heat from 0 to 1. Girlfriend try beginning exactly in the middle and guess 12. Jordan tells you her guess is high, for this reason you understand the prize is what on the number line between 0 and 12. You guess in the center again: 14. Jordan says 14 is quiet high, so you know the answer need to be top top the number line between 0 and 14. Continuing with your strategy, girlfriend guess 18, 316, 532, 964, …. One depiction of this game is presented in figure 1. This seems favor it is taking a lengthy time! will certainly you ever before guess the appropriate number? possibly it would assist if you change your strategy. Or space you doomed to be guessing forever?

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

A new Strategy: Decimal Expansions

Let united state look at these numbers in a different means and think about them as decimals instead. We deserve to turn a fraction into a decimal by splitting the molecule by the denominator. Right here is how it functions for the fraction 716:


For the an initial step the the division, us ask how numerous 16’s room in 70. (Really, we are asking how many 1.6’s are in 7.0, yet this is identical to asking how plenty of 16’s room in 70). Due to the fact that 16 × 4 = 64, we create a 4 over the 0 in 7.0. Then us subtract 64 indigenous 70 and get 6 left over. In this case, 6 is called the remainder.

For the following step, we bring down the next 0 native 7.00. Then us ask how numerous 16’s space in 60. Because 16 × 3 = 48, we write a 3 above the second 0. Next, we subtract 48 from 60 to acquire a remainder that 12.

We continue this process, bringing down zeros after every remainder and asking how many 16’s are in the resulting number. After we have actually done this four times, we get a remainder of 0, which has actually zero 16’s in it. At this point, we room done through our long division and we deserve to say that 716=0.4375. If you space playing the guess-the-number game, you can arrive at this decimal variation of 716 in several short steps. The table listed below shows a possible way this could happen. In the table, H way your guess was too high and also L means your guess: v was as well low.


Because the decimal for the number 716 ends, you can get the exact number by guessing one digit at a time in the decimal. Walk this happen for every fractions? Let us look at the decimal for 322.


Following the same department process, we obtain a 1 on height with a remainder of 8, a 3 on peak with a remainder that 14, a 6 on top with a remainder the 8, a 3 on height with a remainder that 14 … but wait! we have already seen these remainders, and we understand that the next number on height is a 6 with a remainder that 14 again. As we continue to divide, the two repeating remainders the 8 and also 14 give us repeating 3′s and 6′s in the decimal growth for 322. This way that if you try to guess: v the number 322 one decimal location at a time, you will be guessing forever!

Rational Numbers

All that the numbers us have taken into consideration so far are called rational numbers. A rational number is any kind of number the we can write as a fraction ab of 2 integers (whole numbers or their negatives), a and b. This method that 25 is a rational number since 2 and also 5 space integers. Also, 3 is a reasonable number since it have the right to be created as 3=31 and also 4.5 is a rational number due to the fact that it deserve to be written as 4.5=92. Even if we perform not write 3 and 4.5 together fractions, they room rational numbers because we have the right to write a portion that is same to each.

We have seen that part rational numbers, such together 716, have actually decimal expansions the end. We speak to these numbers terminating decimals. Various other rational numbers, such as 322, have decimal expand that save going forever. However we do recognize that also the decimal expand that do not terminate repeat, so we call them repeating decimals.

For any kind of rational number ab, the just remainders we can gain when we compute the decimal room the numbers 0, 1, 2, 3, …, b − 2, b − 1. Because that example, when we were an altering 322 into a decimal, the only alternatives we had actually for remainders to be 0, 1, 2, 3, …, 20, 21. Because there are only a finite number of remainders, the remainders must start come repeat eventually. This is true for every fractions who decimals perform not terminate. Even though over there is a repeating sample to the decimals for these fractions, we will never ever guess the precise number in the guessing video game if we are guessing one decimal ar at a time due to the fact that the decimal goes on forever. We cannot speak infinitely countless digits!

We deserve to go in the turning back direction and readjust decimals come fractions, too! when we have a end decimal expansion, such as 4.132, us can adjust this to a fraction using place 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 much more work to find its equivalent fraction. Because that example, think about 0.353535…. Speak to this number A. The repeating section 35 has actually two digits, so we multiply A by 100 to relocate the decimal over 2 places. This gives 100A = 35.353535…. An alert that every the decimal locations in A and also 100A complement up. We subtract A native 100A to obtain 99A. When we subtract the decimals, the 0.353535… is the exact same for both and is eliminated in the difference. Therefore, we are left through only entirety numbers!


We have 99A = 35, so when we divide by 99, we get A=3599. For any type of repeating decimal, we can use the same process to discover the corresponding fraction. We multiply by 10, 100, 1000, or everything is necessary to move the decimal suggest over far sufficient so the the decimal digits line up. Then we subtract and use the result to discover the equivalent fraction. This means that every repeating decimal is a reasonable number!

Irrational Numbers

What if we have actually a decimal growth that does no end, but the digits execute not repeat? for example, look in ~ 0.101001000100001…. In this number, we rise the variety of 0s in between each pair the 1s, very first having one 0 between, then 2 0s, then 3 0s, etc. This cannot be a rational number since we understand the decimals because that rational numbers one of two people terminate or repeat. This is an instance of an irrational number. One irrational number is any type of number that we have the right to put top top a number line the cannot be created as a fraction of whole numbers. You have actually probably heard around the famed irrational number π = 3.14159…, which offers the proportion of a circle’s circumference to its diameter. When this is a ratio, at least one that the circumference or diameter is no an integer, therefore π is no a rational number. One more irrational number is 2=1.41421…, i beg your pardon is the length of the diagonal line of a square whose sides are length 1.

Going back to our game, all irrational and also rational numbers together fill up our number line between 0 and also 1. Suppose your girlfriend Jordan could pick any number between 0 and also 1 and chose one irrational number for you to guess. You would certainly likely have actually a really hard time guessing the number exactly! just like with the repeating decimal expansion of 322, you can not say infinitely many digits, for this reason this game seems an extremely unfair.

Let us change the game so you can win! Jordan chooses 3 things: a number for you come guess, a variety of numbers in which that number lies, and how close your guess needs to be. Through these new rules, Jordan choose the number π and also tells you “I’m reasoning of a number between 2 and 10. View if you deserve to guess in ~ 0.01 of mine number.” In this situation, the game can go prefer this:


In this brand-new version that the game, even if Jordan alters how nearby you need to guess, friend can always eventually get within that distance of π. You simply need to acquire the entirety number component and a certain variety of decimal places correct. For example, to be in ~ 0.1 that π, you only need to obtain the first decimal place correct. To be in ~ 0.01 the π, you need to acquire the an initial two decimal areas correct. To be in ~ 0.001 the π, you require to get the first three decimal places correct. No matter just how close her guess must be, you have the right to win this new game through guessing one decimal location at a time till you have enough decimal places.

As we observed earlier, every decimal that terminates is a reasonable number. If we use this process of getting closer and also closer to an irrational number by guessing more and an ext decimal places of the number, we can acquire a reasonable number that is together close together we prefer to our score irrational number. In our game, this way that no issue what irrational number Jordan chooses nor just how close you have to guess, girlfriend can constantly find a rational number the will accomplish the requirements. In this game, you can always win!


The reason this happens is that the rational numbers space dense in the real numbers. This means that between any type of 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 kind of irrational number with a rational number. Approximating an irrational number through a rational number is what you room doing in the new game as soon as Jordan picks an irrational number.

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But why would certainly you ever before need to approximate an irrational number with a rational number? expect you are building a wooden frame for a triangular garden bed in the shape of fifty percent the square in number 2. You require to reduced a piece of lumber that is 2 feet long. Exactly how will girlfriend measure the length? because 2 is one irrational number, friend cannot usage your tape measure up to measure up it exactly! Instead, you will pick a reasonable number that approximates 2. You can choose the variety of decimal number to include in your development in bespeak to acquire the item of hardwood as close in length as you desire to 2, the same way you determined your rational number to be together close together Jordan wanted you to gain in the game.