Roxbury Math Epsilon Club Lecture2

Lecture 2

Roxbury Math Epsilon Club

Let’s review what we did last time with yet another example. We all know that we need to learn our multiplication table… But how big is it, how many multiplications do we actually need to memorize? At first pass, it would seem that we need to memorize entries. But that seems too many because we would be double-counting most of the entries. After all:

, so we have a bit less… but how much less.

Sidebar: Why is ? Answer: Commutative Law. I will talk about this a bit later today Let’s try to count how many multiplications we need to learn simply by adding one number at a time. We start with “lonely” 1. All we need to learn is: . (1 multiplication) Now we add “2”. Well, we now need to memorize: and Let’s add “3”. New items to memorize are: , and Are you seeing a pattern? Every time we add a new number, the number of additional items we need to memorize is the same as the number we’re adding. We had 1 for “1”, 2 for “2”, 3 for “3” and so on. We also know that we need to add all the numbers from “1” to “9”. Let’s be real mathematicians and call this unknown number “X”.

But wait a minute… we studies this formula in Lecture 1. This is just a sum of numbers from “1” to “9”. This is easy… we already know how to calculate this (and we did the exact same calculation for “Magic Square” problem as well).

So, what if we wanted to memorize all the 1-digit and 2-digit multiplications? How many multiplications would that be? The other way of asking this question is:

Lecture 2

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Notice, that in this case, I do not include 100, since we only want 2 digits multiplications, excluding 100.

Lecture 2

Roxbury Math Epsilon Club

Let’s talk about another subject…

FRACTIONS.

I know that you’ve already studies this before and studying it now. I just wanted to show how various forms of fractions are related to one another and why there are so many of them in the first place. Fraction is just a number and various representations is how we “write” this number down. No matter how we “write it” - the value of the number does not change. Let’s illustrate with example:

7 1 2    0.2  20% 35 5 10 Because there is an equal sign between all these representations, I am allowed to substitute whichever one is the most convenient for me when I am figuring out my expression. 1 30  20%  30   6 5 7 73%   73%  20%  93% 35 2 0.35   0.35  0.2  0.07 10 7 35  0.2  35   7 35 In each of the problems, I chose a way which makes it easiest for me to solve my particular problem. Most of the times, I will not necessarily have the “best” representation ahead of time and will have to convert to it as necessary. Let’s look at another example:

1 21 8 21 29     5 40 40 40 40 I did not have the original fraction represented in the way which I needed, so I had to find the necessary representation in order to make my calculation simple (make both fractions have the same denominator). The key is to make sure that the substituted fraction has the same value as before.

1 1 1 8 1 8 8  1     5 5 5 8 5  8 40

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The reason that you can multiply numerator and denominator by the same number and end up with the same fraction is that you’re really multiplying the original fraction by “1” and therefore not changing its value. So why are there so many ways to represent fractions? The following table summarizes all three fraction forms and their advantages and disadvantages.

Representation

Main Purpose

Advanta ge

Disadvantage

The most natural way to represent a fraction

Accurate compact representation . Easy to multiply and divide Easy to add, subtract, multiply, divide, compare

Hard to add, subtract, compare

Imply denominator by number of digits after decimal period Make denominator equal to 100.

Really easy to compare

Only fractions with certain denominators can be represented accurately. General fraction may require infinite number of digits In addition to being decimal, numerator may also become a fraction as well

Let’s see an example: 1 333... 33.333... 1   0.333....   33.333...%  33 % 3 1000... 100 3 So, every decimal can be represented as a normal fraction, but even a simple normal fraction may require infinite number of decimal digits to represent. In practice, we usually use several decimal digits and round off the remaining ones. Then,

1  0.333...  0.33  33% 3 Notice that,

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2  0.666...  0.67  67%  66% 3 Make sure that if you’re going to round off a decimal, use the proper round-up/rounddown rule. Sidebar: Notice that there is a trade-off between normal fractions and decimals. Fractions are exact but harder to deal with. Decimals are sometimes not as exact, but easier to deal with. This is very common in math --- instead of solving a harder, general problem, we find easier, approximate solution. Who wants to solve tough problems, anyway! Here are a few more ideas on how to deal with various fraction problems and such: 1) Factorization of numerator and denominator. Let’s demonstrate with an example:

14  69  2  7    3  23 2  7  3  23 2  3    6 7  23 7  23 1 7  23 Let’s also look at what not to do: 567 56 7  7 56 56 Above would be correct, if

, but of course, that’s not the case. Instead:

2) Take a percent of a number. This is exactly the same as multiplying that number by a percent (which in turn is the same as multiplying a number by the equivalent fraction).

600  20%  600 

20 600   20  6  20  120 100 100

3) Any whole number can be easily represented as a fraction:

621 

621 1

For that matter, any division is the same as a fraction and any fraction is a division. In math, this is called one-to-one mapping. So

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25  5 

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25 25  2 50    50  10  5 5 5  2 10

4) We can actually have fractional numerator and denominator. Even though, it looks strange, as long as we approach it systematically, it will not be any different from our usual calculations. Two main rules to remember are: a) In order to multiply fractions, multiply their numerators and their denominators b) Dividing by a fraction is the same as multiplying by an “inverse” fraction. Here is an example:  1   1  2      4   4  2      3  2  3 6 3    4  1    2   30  2  30  20 3  3    4 To solve the second problem, just use the answer from the first one. Don’t start all over from the beginning!!! 5) Any number can be a numerator of a fraction (including zero)… and just about any number can be a denominator in a fraction… There is only one exception. Which number is always illegal as a denominator? ZERO

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Let’s also talk a bit more about percents. The main reason people use them is to easily compare “things”. Say, for example we have two schools: A and B. School A has 50 boys and 50 girls, while school B has 40 boys and 20 girls. Question is which school has more boys. The answer is… it depends. If you count by actual number of boys, of course school A has 10 more boys as compared to school B. On the other hand, let’s calculate the number of boys in each school as a percent of the school population. While school A has: , school B has . Then as a percent of the school population, school B actually has more boys as compared to school A. While the ratio of boys to girls in school A is: , the same ratio in school B is: If someone told you that some school C had 30% population of boys, even without knowing any exact numbers, we would know right away that school C had many more girls as compared to boys. If school D had 49% boys, we would know that school D had just slightly fewer boys than girls as compared to school A (which has 50% boys). Again, we would not need any other numbers beside the percent. The part of math which studies these types of questions is call: Statistics.

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Mathematical Vocabulary Word-Of-The-Day: Commutative Law. It’s very simple: This law is a property of an operation… in this case “+”. To what other operations to this law apply… and to which operations it does it not apply. Mathematical Vocabulary Greek-Letter-Of-The-Day: Omega. Lower case omega looks like this:

, while upper case looks like this:

In math and physics, this letter usually represents the speed of rotation… but that’s for another day.

Lecture 2

Roxbury Math Epsilon Club

Homework… I would like everybody to think of really clever solutions to these very formidable math problems… 1) Evaluate the following fraction:   1    2        3    4      ?   5    6        7    8      Hint: first evaluate the numerator, then the denominator 2) Evaluate the following expression:

1 2 3 99    ...  ? 2 3 4 100 Hint: you do not want to do 99 multiplications in the numerator and 99 multiplications in the denominator!!! 3) Even though, factoring individual digits in not correct, there is one very nice exception to this rule where it is correct, and in fact it is done all the time. Can you think of when this exception takes place and what specific digit it involves? 4) If you had to memorize only 2-digit multiplications, excluding any 2-digis by 1-digit or 1-digit by 1-digit multiplications, how many multiplications would you need to memorize? We will discuss all the solutions next time…