Repeating Decimals

The Cool Thing about Repeating Decimals

Get out your calculator and type in 331/999 and press enter. What do you get? Something along the lines of 0.331331331…? Now type in 21/9999 and press enter. What do you get? 0.002100210021…? This cool trick works with any fraction less than 1 with all nines in the denominator, which are always of the form 10A – 1, for some number A. Here’s a proof:

  • Say x = 0.A1A2A3…An…… (Where x is a repeating decimal, and A1A2A3…An is the string of numbers that is repeated in the decimal, and n is the length of the string)
  • If we multiply both sides by 10n, then we would get: x10n = A1A2A3…An.A1A2A3…An…, because the decimal point is moved n times to the right
  • x10n = A1A2A3…An.A1A2A3…An… is the same thing as A1A2A3…An + 0.A1A2A3…An, and since 0.A1A2A3…An = x, then x10n = A1A2A3…An + x
  • By subtracting both sides of the equation by x, you would get x10n – x = A1A2A3…An, or x(10n – 1) = A1A2A3…An
  • By dividing both sides by 10n – 1, you would get… x = (A1A2A3…An)/(10n – 1)
  • Therefore, by substituting 0.A1A2A3…An… for x, you would get…
  • A1A2A3…An… = (A1A2A3…An)/(10n – 1), where A1A2A3…An is a string of numbers of length n, and 10n – 1 is a string of n 9s, for example 103 – 1 = 1000 – 1 = 999

This proof is a generalization of a trick you learn in middle school. Say you want to convert 0.21… into a fraction. You would set that equal to x to get x = 0.21… and multiply both sides by 100 to get 100x = 21.21…, which is 100x = 21 + 0.21…, or 100x = 21 + x. Next you would subtract x from both sides to get 99x = 21, and x = 21/99. Don’t forget to simplify J.

Note: If you want to do something like 7/99999, add 0s in front of the 7 to make the numerator and denominator have the same number of digits to get 00007/99999, and the decimal of that would be .000070000700007…

Back-to-School Success

Tips for Students and Parents

And just like that, another summer is over and a new school year begins! Here are some tips for both parents and students to work together to ensure a successful school year.

Students

1.  Set goals: Write them out clearly and display them somewhere that you see them everyday

                ex: I will complete my HW before I watch TV

2.  Get organized: This includes finding a way to organize papers going back and forth from subject to subject.  How are you going to know what your assignments are and when they are due? 

3. Plan: What HW, tests, and quizzes do you have this week?  How will you prepare for them?  Make sure you plan out your study time.

4. Practice: This is how you learn!  Make time each day to practice.

5. Get Help: Are you not understanding what you are supposed to be learning?  Ask!  Get help!  Go to your teacher, parent, and of course, MaThCliX!  That is what we are here for.

Parents

1.  Make sure that you know how to communicate with your student’s teacher.  Know when conferences are and plan to have a presence and be proactive in your student’s academics.

2. Check grades!  Even if your student is old enough to check their own grades, it never hurts to have a parent checking, too.  Know when progress reports and report cards are due.  If you see grades dropping, intervene quickly!

3.  Make sure your student is doing the success tips for students.  Ask them how they are doing each one.

4.  Find out about what student’s are learning each week so that you can help or get help, as needed.  Find out about tutorials, teacher websites, and recommended resources.

5. Bring your student to MaThCliX! 

Join Math Team

You may be discouraged from joining the math team because you don’t think you’re not good at it, but like many other things in life, you can get better at math by working hard at it and with the proper attitude. All you need is a love of math, the desire to make yourself better, and a winning attitude. If you don’t like math, then maybe joining math team can change your opinion on math. At math competitions, you commonly get problems that require you to think outside the box and think of creative solutions, unlike traditional high school math. If you are interested or just want to try it out, talk to your school’s math team coach and ask him or her about joining the team. He or she would be more than willing to help you out. If your school doesn’t have a team and you want to start one, take initiative and find a teacher who is willing to sponsor and coach a math team. There are lots of math competitions in Georgia every year, where you get a chance at showing off your math skills, finding opportunities to improve, and simply having fun. If you see a problem you don’t understand, don’t feel discouraged. If you are willing to learn the tricks, you will get better and better at it, and eventually, if you do consistently well at math competitions, you can make it on one of Georgia’s state ARML teams, where the best are selected from all over Georgia to compete in a competition involving teams from all over the US and even from other countries, like Colombia and China. You get to interact with kids who love math just like you from all over Georgia and stay at a dorm at the University of Georgia to #DoMaTh and/or socialize with the kids and compete for a chance for fun and international glory.

Mathematical Induction

Why I Don’t Like Mathematical Induction

Everybody has opinions, and you can even have opinions about math. I, for one, don’t particularly like the method of mathematical induction. Why? Because it doesn’t completely tell you “why”. For example, you can use mathematical induction to prove that 1 + 2 + 3 + 4 + … + n = n(n+1)/2. Great, we know that the formula works, but where does the formula come from? A much more elegant method to prove that formula is derivation. There are many ways to derive that formula, and one method is by writing 1 + 2 + 3 + 4 + … + n forwards and backwards and by adding them together twice. That way, you can see how twice the sum would be n(n+1) (see the visual), so the sum would be n(n+1)/2. By deriving the formula, you see why the formula is the way it is, and you’ll be able to connect the formula to the nature of the series. Also, mathematical induction is not a method to make equations, as it can only be used if the equation is given. With induction, you may prove old equations, but with derivation, you’ll be able to make new equations.

Pythagorean Triplets

Cool Thing About Pythagorean Triplets

There is a cool property of all odd numbers except one, they can all form Pythagorean triplets. A Pythagorean triplet is any set of positive integers a, b, and c that satisfy the equation a2 + b2 = c2. For example, a= 3, b= 4, and c= 5 is a Pythagorean triplet because 32 + 42 = 52. If a is any odd number except 1, it can make a Pythagorean Triplet of the form a= a, b= (a2 – 1)/2, and c= (a2 – 1)/2 + 1, or b + 1. You can prove this by substituting b and c with b= (a2 – 1)/2 and c= (a2 – 1)/2 + 1 in the equation a2 + b2 = c2. “a” cannot be even for this case because if a were even, (a2 -1)/2 would not be a whole number, and a cannot be one, because then, b would be zero. For all other odd numbers, this would work, and Pythagorean Triplets of this form include…

a= 5, b= 12, c= 13

a= 7, b= 24, c= 25

a= 9, b= 40, c= 41

Setting Goals

Setting Goals for the Semester

As you progress through a semester, it is important to continuously assess your progress and set new goals for yourself. This is an important thing to do because it makes you more aware of how you are performing and helps you to create a focused path for the remainder of the term. Furthermore, setting goals helps you to better organize your limited time and resources available as well as improve your motivation

Assess Your Progress:

The first step to goal-setting is to assess your current progress. This includes knowing the grades you have received as well as the study habits you used to get them. Make a note of whether you have been doing things like reading the textbook, taking notes in class, completing your homework or thoroughly reviewing before tests. It is also important to assess how well you understand the course material as well as if your grades match your perceived understanding.

Decide What You Want from the Class:

Before setting defined goals, it is important to think about what you want to get out of the class. Do you want a good grade to raise your GPA? Do you want a strong foundation on a topic that will be important later in your curriculum? Do you want to be more knowledgeable about a subject that interests you? Knowing what you specifically want from a course will give you the motivation to keep focused and continue to work hard.

Setting Goals:

Now that you know how you are doing in your courses and what you want to gain from them, it is time to set goals for yourself.

Set goals for things you would like to see occur. For example, maintaining or improving your grades, study habits or your understanding of certain topics. It is important that your goals are both reasonable and specific to keep you motivated and focused.

In addition to setting goals, you need to create a plan to help you reach them. To get the grades that you want, you must understand what study habits you will need to have. Also, you must assess if you need extra help and find out what resources are available to you. The internet may have additional explanations and examples of topics. Additionally, your teacher may be able to give you additional help outside of class. Don’t forget that MaThCliX is also a great resource!

Once you’ve set goals and created a plan for yourself, you will be on track for a focused and productive term. Following the ideas discussed above will help you stay motivated while providing with the skills and resources to get the most out the courses you are taking.