ns am an associate Professor of mathematics at Juniata university in Huntingdon, PA. I have actually done math research in geometric combinatorics and also linear algebra. I am also really interested in how civilization learn mathematics and look for avenues to an obstacle students and teachers to know math an ext deeply. *camenga

ns am an associate Professor of mathematics at Houghton university in western brand-new York state. My recent research is in direct algebra and graph theory. Ns am an analyst through training, so I have actually an abiding interest in the actual numbers and their properties. Ns really prefer helping world learn to think prefer mathematicians, every little thing their backgrounds might be, and helping teachers help their students do deep connections.

Hello! My surname is Ivy and I to be in saturday grade. I reap doing math and drawing and also am top top my school MathCounts team. I play piano and violin. Ns like huge microbes.


Jonathan is 13 years old and going into eighth grade. He enjoys all music except pop. And of course, math is among his favourite subjects. He plays violin in a youth orchestra and oboe in college band.

Hello! i am Samantha Singh, a climbing seventh grader in ~ Wayzata main Middle School and second-year student in the UofM’s Talented Youth math Program. I am an admit of the college of Notre Dame’s NDignite Program, a routine that disclosure leadership among high-achieving students. Additionally, I occupational on the editorial plank for KidSpirit magazine and participate in an initial Lego League, receiving nomination for the an international Innovation Award. At some point I expect to architecture medical devices which will save numerous human lives.


Legend has it the the an initial person in ancient Greece who found that there room numbers the cannot be written as fractions to be thrown overboard indigenous a ship. Centuries later, when we consistently use numbers the cannot be written as fractions, those numbers that have the right to be created as fountain remain an effective tools. What renders fractions so special? We check out how we have the right to recognize the decimal representation of fractions and how fractions have the right to be offered to approximate any kind of real number as closely as we wish.

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

But after ~ class, you again run right into Jordan, who has actually apparently been thinking around ways come stump you: why rod to positive numbers? What if you additionally allow an unfavorable numbers? “Now i am thinking of a number between an unfavorable 100 and also 100,” Jordan claims gleefully. You decision to take it the bait, and you quickly find that this walk not change the video game much. Girlfriend guess, and 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 too low and also −31 is as well high, climate you know the price is −32. Yet then you realize: over there is naught special around −100 and 100! If you start with a number between −1000 and 1000, you know you will at some point guess the exactly number also if it takes a couple of more guesses. Friend march off to your second class victoriously, confident the you will certainly be ready for Jordan’s next challenge.

However, during that class, girlfriend realize that you have actually been presume Jordan will always pick one integer. What if fractions room allowed? expect Jordan choose a number between 0 and 1, for example 322. You need to guess a number somewhere along the number heat from 0 come 1. Friend try beginning exactly in the middle and also guess 12. Jordan tells you her guess is high, for this reason you recognize the prize is somewhere on the number line between 0 and also 12. Girlfriend guess in the center again: 14. Jordan claims 14 is quiet high, for this reason you understand the answer must be top top the number line between 0 and also 14. Continuing with her strategy, girlfriend guess 18, 316, 532, 964, …. One representation of this video game is presented in figure 1. This seems favor it is acquisition a lengthy time! will you ever guess the appropriate number? probably it would help if you readjust your strategy. Or space you doomed to it is in guessing forever?

Figure 1 - A number guessing game.Your girlfriend Jordan asks you to guess a number in between 0 and 1. Through each guess, girlfriend halve the variety where Jordan’s number can be. The dot at the end of every line segment is your guess. The place of the number you room trying come guess, 322, is marked by the vertical black line segment.

A brand-new Strategy: Decimal Expansions

Let us look at this numbers in a different means and think about them as decimals instead. We have the right to turn a portion into a decimal by dividing the numerator by the denominator. Below is how it works for the fraction 716:


For the first step of the division, we ask how countless 16’s room in 70. (Really, we are asking how plenty of 1.6’s room in 7.0, but this is indistinguishable to questioning how many 16’s room in 70). Because 16 × 4 = 64, we write a 4 above the 0 in 7.0. Then we subtract 64 native 70 and get 6 left over. In this case, 6 is dubbed the remainder.

For the following step, we lug down the following 0 native 7.00. Then us ask how numerous 16’s space in 60. Since 16 × 3 = 48, we write a 3 above the 2nd 0. Next, we subtract 48 native 60 to acquire a remainder of 12.

We proceed this process, bringing down zeros after every remainder and asking how plenty of 16’s room in the result number. ~ we have done this four times, we obtain a remainder of 0, which has actually zero 16’s in it. At this point, we space done v our long division and we have the right to say the 716=0.4375. If you are playing the guess-the-number game, you can arrive in ~ this decimal variation of 716 in several brief steps. The table below shows a possible method this could happen. In the table, H means your guess was also high and also L means your guess: v was also 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. Go this occur for all fractions? Let us look in ~ the decimal for 322.


Following the same division process, we get a 1 on height with a remainder that 8, a 3 on optimal with a remainder that 14, a 6 on optimal with a remainder that 8, a 3 on height with a remainder of 14 … however wait! us have currently seen these remainders, and we understand that the next number on optimal is a 6 through a remainder that 14 again. As we continue to divide, the two repeating remainders the 8 and 14 give us repeating 3′s and 6′s in the decimal growth for 322. This way that if you try to assumption: v the number 322 one decimal ar at a time, you will certainly be guessing forever!

Rational Numbers

All the the numbers us have considered so much are referred to as rational numbers. A rational number is any kind of number the we can write together a fraction ab of 2 integers (whole numbers or their negatives), a and also b. This method that 25 is a rational number since 2 and 5 are integers. Also, 3 is a rational number due to the fact that it can be composed as 3=31 and also 4.5 is a reasonable number due to the fact that it have the right to be created as 4.5=92. Even if we do not create 3 and 4.5 together fractions, they room rational numbers because we have the right to write a portion that is equal to each.

We have actually seen that part rational numbers, such together 716, have decimal expansions that end. We contact these number terminating decimals. Various other rational numbers, such together 322, have actually decimal extend that store going forever. But we do understand that even the decimal expansions that perform 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 space the numbers 0, 1, 2, 3, …, b − 2, b − 1. For example, as soon as we were transforming 322 into a decimal, the only choices we had for remainders to be 0, 1, 2, 3, …, 20, 21. Since there are just a finite variety of remainders, the remainders have to start to repeat eventually. This is true for all fractions whose decimals execute not terminate. Also though there is a repeating pattern to the decimals because that these fractions, we will never guess the precise number in the guessing game if we room guessing one decimal place at a time because the decimal walk on forever. We cannot say infinitely countless digits!

We deserve to go in the reverse direction and adjust decimals come fractions, too! as soon as we have a end decimal expansion, such as 4.132, us can change this to a fraction using place value. The 2 the 4.132 is in the thousandths place, so 4.132=41321000. If we are starting with a repeating decimal, we have to do a bit an ext work to find its equivalent fraction. Because that example, think about 0.353535…. Contact this number A. The repeating part 35 has actually two digits, so us multiply A through 100 to relocate the decimal over two places. This gives 100A = 35.353535…. Notification that all the decimal places in A and 100A enhance up. We subtract A indigenous 100A to gain 99A. When we subtract the decimals, the 0.353535… is the very same for both and is eliminated in the difference. Therefore, we are left with only totality numbers!


We have actually 99A = 35, so when we division by 99, we get A=3599. For any kind of repeating decimal, we have the right to use the same procedure to find the corresponding fraction. Us multiply by 10, 100, 1000, or every little thing is necessary to relocate the decimal suggest over far enough so the the decimal digits heat up. Then we subtract and use the an outcome to uncover 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, yet the digits execute not repeat? because that example, look at 0.101001000100001…. In this number, we increase the variety of 0s between each pair the 1s, first having one 0 between, then 2 0s, then 3 0s, etc. This cannot be a reasonable number since we understand the decimals because that rational numbers either terminate or repeat. This is an instance of an irrational number. One irrational number is any number that we can put on a number line that cannot be written as a portion of totality numbers. You have probably heard about the famed irrational number π = 3.14159…, which gives the ratio of a circle’s circumference to its diameter. When this is a ratio, at the very least one that the circumference or diameter is no an integer, for this reason π is no a rational number. An additional irrational number is 2=1.41421…, i beg your pardon is the length of the diagonal line of a square whose political parties are length 1.

Going back to our game, all irrational and also rational numbers with each other fill up our number line between 0 and 1. Expect your friend Jordan could pick any kind of number in between 0 and also 1 and also chose one irrational number for you to guess. You would likely have actually a an extremely hard time guessing the number exactly! as with with the repeating decimal growth of 322, you can not say infinitely many digits, so this game seems really unfair.

Let us change the video game so you have the right to win! Jordan chooses 3 things: a number for you to guess, a selection of numbers in which that number lies, and also how close her guess needs to be. Through these brand-new rules, Jordan chooses the number π and also tells you “I’m reasoning of a number between 2 and 10. View if you have the right to guess in ~ 0.01 of my number.” In this situation, the game could go choose this:


In this new version of the game, also if Jordan transforms how close you have to guess, girlfriend can constantly eventually acquire within that street of π. You simply need to obtain the totality number component and a certain number of decimal areas correct. For example, come be in ~ 0.1 the π, girlfriend only require to gain the very first decimal location correct. To be within 0.01 of π, you need to obtain the very first two decimal areas correct. To be in ~ 0.001 the π, you require to obtain the first three decimal places correct. No matter exactly how close your guess need to be, you can win this new game by guessing one decimal location at a time until you have sufficient decimal places.

As we saw earlier, every decimal that terminates is a rational number. If we use this procedure of acquiring closer and also closer come an irrational number by guessing more and much more decimal places of the number, us can acquire a rational number that is as close together we favor to our goal irrational number. In ours game, this method that no issue what irrational number Jordan choose nor just how close you have to guess, girlfriend can constantly find a rational number the will meet the requirements. In this game, girlfriend can always win!


The reason this wake up is the the reasonable numbers are dense in the real numbers. This means that between any kind of two different real numbers, us can constantly find a rational number. Because real numbers have this property, we can approximate any kind of irrational number v a rational number. Approximating one irrational number v a reasonable number is what you space doing in the brand-new game as soon as Jordan picks an irrational number.

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But why would you ever before need to almost right an irrational number v a rational number? suppose you are building a wooden framework for a triangle garden bed in the shape of fifty percent the square in number 2. You require to cut a piece of wood that is 2 feet long. Exactly how will friend measure the length? since 2 is one irrational number, friend cannot usage your tape measure up to measure it exactly! Instead, you will pick a rational number that approximates 2. Girlfriend can pick the variety of decimal digits to encompass in your development in order to acquire the item of lumber as near in length as you want to 2, the same method you chose your reasonable number come be together close as Jordan wanted you to get in the game.