Course:MATH110/Archive/2010-2011/003/Groups/Group 12/Mathematical Writing

Introduction

Mathematical Writing is the process of constructing a series of logical steps that follow from each other to prove a solution to a mathematical concept or question. Think of Mathematical Writing as a convincing argument on how and why the solutions to a concept or a question is what it is. If another person were to read one's mathematical argument, they should be able to understand how one derived a solution without further explanation. Of course, there are certain basic assumption or skills that need to be mastered for certain solutions. In essence, a mathematical argument is self contained, containing all the necessary information for one to understand a problem. Likewise, Mathematical Writing ensures accuracy because it allows yourself and others to determine if there were any mistakes within your argument. Problems with your argument should be evident upon reviewing.

In the realm of Philosophy, the skill of writing an argument is paramount to the success of a Philosopher. Deriving solutions to problems can be easy, but deriving solutions that follow logically with supportive reasoning can be difficult because it really tests one if he or she truly understands the problem at hand. The skill of arguing is a universal trait that everyone needs to learn, no matter what discipline he or she is in. A convincing and sound argument will always appear more believable and persuasive than just an answer. It is because of these reasons that Mathematical Writing is an essential skill to master. While not all of us will be pursuing mathematics as a discipline, all of us will have to make a convincing case about ourselves or to others throughout our lives. Mastering Mathematical Writing will prime us for future courses and enable us to become presenters.

If you are a more concrete thinker, mastering mathematical writing is essential to your success in mathematics. You truly understand a concept if you are able to explain it. Explaining solutions instead of recognizing and reacting to patterns also helps to reinforce the concepts that you have learned.

If you are more stubborn to why you should master mathematical writing, then quite frankly, you'll need to master this skill if you want to do well on your final exam!

Techniques and Examples

We now know that Mathematical Writing is important, but how exactly do we go about in actually writing a mathematical argument? Listed below will be a few techniques to help us write a convincing argument.

Group 12's Technique

First we will explain how our group does Mathematical Writing because we have found that our technique has given us success for the assignments. Our technique logically applies previous knowledge and directly proves what we want to show. Think of it as writing out a solution with all the steps to a problem, but with a clearer structure and supportive reasoning.

• Define the problem and state all your assumptions: It can be argued that this first step is the most crucial step in Mathematical Writing. It is important to state our understanding of the problem and state any assumptions we have made because the logical steps that follow will be based on our definitions and assumptions. It also allows the reviewer to know what we are thinking and understand why we made certain conclusions. This step also makes clear to us how exactly we understand the problem and how we should approach the problem at hand. If possible, state the final conclusion that you want to prove.
• State all the given premises, that is, state all the given information as a numbered list: A numbered list is important because it makes the flow of our arguments clear and understandable. It also allows us to quickly cite the exact steps we needed to make a certain conclusion for further steps.
• Derive conclusions from the listed premises: Using the given premises, it is now the time to derive conclusions from these premises. As we are explaining each logical conclusion, we must make sure to state what premises were involved in making that conclusion. An concise explanation is also needed to explain how we made those conclusions.
• Conclusion: Assuming you properly defined the problem at hand, you should recognize when you reached the final conclusion. You should also have an idea whether your answer is correct based on the previous conclusions you have made. If you are unsure that your final conclusion is correct, it is now the time to review your previous conclusions to see if any logical mistakes were made. If you think that your conclusions flows logically, state your final conclusion with any of the premises that you have derived from above alongside with a concise explanation. Philosophically, the notation of QED should mark the end of your argument. QED means "which was to be shown or proven".

Example: Prove that the sum of any two even integers is even.

Define the problem and state all your assumptions: We know by definition, that integers are whole numbers and even numbers are numbers that are divisible by 2. Let the first even integer be represented by an X and the second even integer be represented by an Y.

State all the given premises:

• 1) X + Y = Even Number

Derive conclusions and state your final conclusion:

• 2) 2a + 2b = Even Number (1. Substitute X with 2a and Y with 2b because by definition of a even number we know that an even number is divisible by 2, so any number multiplied by 2 will be even.)
• 3) 2(a + b) = Even Number (2. By distribution of integers.)
• 4) It is clear from (3) that it has a factor of 2, thus making it a even number. Therefore, the sum of any two even integers is even.
• QED

Mathematical Induction

This form of reasoning allows us to prove that a statement holds for any value. Mathematical induction is particularly helpful in proving concepts that want us to accomodate for infinite cases that cannot be easily proved or impossible to prove directly (e.g., our technique proves statements directly).

To illustrate what mathematical induction is all about, think about a row of infinite dominoes. We know that if we push one of the dominoes over, that the next domino in the row will fall over. You can see by pushing one domino over that it will create a chain of events. Using this illustration, we can form a basic framework for mathematical induction:

• 1) Base Case: We knock over the first domino
• 2) We prove that if we knock over any domino, that the next one will fall over.
• 3) We can conclude all dominos will fall since (1) and (2) create a chain of events given that we can plug any domino into (2).

Example: Prove that $1+2+3+...+n={\frac {n(n+1)}{2}}$

• 1) Premise: $1+2+3+...+n={\frac {n(n+1)}{2}}$
  
  
• 2) Base Case: Let n = 1; 1 = 1(1 + 1). 1 = 1; therefore our base case holds. (n = 1 because it is the first case that we can show. When showing for the base case, you always want to use the first case.)
  
  
• 3) Inductive Hypothesis: Substitute k for n. Which gives us: $1+2+3+...+k={\frac {k(k+1)}{2}}$
• 4) Inductive Step: Using our domino example, we want to show that the next domino will fall over: $1+2+3+...+(k+1)=(k+1){\frac {((k+1)+1)}{2}}$
  
  
• 5) We now must show that (3) = (4).
  
  
• 6) $1+2+3+...+k+(k+1)={\frac {(k+1)(k+2)}{2}}$ (The sum of the numbers from 1 to k + 1)
  
  
• 7) The underlined part looks familiar, doesn't it? It is (3)! From (3), we know the underlined part is equal to ${\frac {k(k+1)}{2}}$, so let's substitute that in: ${\frac {k(k+1)}{2}}+(k+1)={\frac {(k+1)(k+2)}{2}}$
  
  
• 8) We are now at the point where we can finally use some algebra to finish this proof: ${\frac {(k^{2}+k)}{2}}+(k+1)={\frac {(k^{2}+3k+2)}{2}}$ (Multiply through)
  
  
• 9) ${\frac {(k^{2}+k)}{2}}+{\frac {2(k+1)}{2}}={\frac {(k^{2}+3k+2)}{2}}$ (Find common denominator)
  
  
• 10) ${\frac {(k^{2}+3k+2)}{2}}={\frac {(k^{2}+3k+2)}{2}}$ (Both sides equal the same, therefore by mathematical induction, $1+2+3+...+n={\frac {n(n+1)}{2}}$
• QED

For another example regarding mathematical induction, you can check out David's proof of the Power Rule.

Problem Sets

As you can see, Mathematical Writing can be very tedious, but you can also appreciate the intricate details that go into a problem solution. Since Mathematical Writing can generate very different answers from many students, you are encouraged to post your solutions at our solutions subpage. Simply state the problem you are attempting to answer with your supplied solution. While many answers can appear different, it will be interesting to see the different solutions that fellow students come up with. This will be a good review for others to see the style differences and what separates good solutions from poor solutions. Because of this fact, this section will not contain any supplied "correct" answers. So please participate! If participation appears to be low, we will post some of our answers to start the discussion.

Questions

• 1) Explain and Compute: $\lim _{x\rightarrow 3}={\frac {5x^{2}-8x-13}{x^{2}-5}}$
  
  
• 2) Explain and Solve: Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s how fast is the area of the spill increasing when the radius is 30 m?
  
  
• 3) Prove that every odd positive integer can be written as the difference of two perfect squares.
  
  
• 4) Prove for all integers $n\geq 1:1+6+11+16+...(5n-4)={\frac {n(5n-3)}{2}}$

Another good source of practice material is from your course textbook. Since all the even questions have the answers, you can easily determine if you have the right answers or not. These questions will provide you with ample practice for mathematical writing and also serve as an excellent review for the upcoming final exam.

References

Listed below are some helpful links that discuss more about Mathematical Writing:

• A Guide to Mathematical Writing.
• How to Write Proofs. (Contains more techniques that are more geared towards logic problems.)