Roxbury Math Epsilon Club Lecture2

8/14/2019 Roxbury Math Epsilon Club Lecture2

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Lecture 2 Roxbury Math Epsilon Club

Lets 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 willtalk about this a bit later today

Lets 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

Lets 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 were 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. Lets 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 theexact 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:


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

excluding 100.


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Lets talk about another subjectFRACTIONS.

I know that youve 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 ofthem 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.

Lets illustrate with example:

2

10

720

535%

10.2

Because there is an equal sign between all these representations, I am allowed tosubstitute whichever one is the most convenient for me when I am figuring out my

expression.

30 30 6%1

520

73% 73 27

30% 9 %

5% 3

0.35 0.3 0.2

105 0. 72 0

35 35 7235

0.7

In each of the problems, I chose a way which makes it easiest for me to solve myparticular 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.

Lets look at another example:

8

40

21 21 29

40 4 0

1

5 0 4

I did not have the original fraction represented in the way which I needed, so I had to findthe 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 8 1 81

5 5 8

1

5 5 8

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 youre 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 MainPurpose

Advantage

Disadvantage

The most natural

way to representa fraction

Accurate

compactrepresentation

. Easy tomultiply anddivide

Hard to add, subtract,

compare

Imply

denominatorbynumber of digits

after decimal

period

Easy to add,

subtract,multiply,

divide,

compare

Only fractions with

certain denominators canbe represented accurately.

General fraction may

require infinite number ofdigits

Makedenominator

equal to 100.

Really easy tocompare

In addition to beingdecimal, numerator may

also become a fraction aswell

Lets see an example:

1 333... 33.333... 10.333.... 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, weusually use several decimal digits and round off the remaining ones. Then,

10.333... 0.33 33%

3

Notice that,


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20.666... 0.67 67% 66%

3

Make sure that if youre going to round off a decimal, use the proper round-up/round-

down rule.

Sidebar: Notice that there is a trade-off between normalfractions and decimals. Fractions are exact but harder to dealwith. Decimals are sometimes not as exact, but easier to dealwith. This is very common in math --- instead of solving aharder, 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. Lets demonstrate with an example:

2 7 3 2314 69 2 7

7 23 7 23

3 23

7 232 3

61

Lets also look at what not to do:

567 56

56

7

567

Above would be correct, if , but of course, thats 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 equivalentfraction).

20 600600 20% 600 20 6 20 120

100 100

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

621621

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 25 2 5025 5 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 4 4 22

3 2 3 6 3

4

1

2230 30 20

3 3

4

To solve the second problem, just use the answer from the first one. Dont start all over

from the beginning!!!

5) Any number can be a numeratorof a fraction (including zero) andjust about any

number can be a denominatorin a fraction There is only one exception. Which

number is always illegal as a denominator?

ZERO


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Lets 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 boysand 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 comparedto school B.

On the other hand, lets calculate the number of boys in each school as apercentofthe 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 withoutknowing 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. Its very simple:

This law is a property of an operation in this case +. To what other operations to thislaw 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 thats for

another day.


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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 niceexception 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

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Roxbury Math Epsilon Club Lecture2