positive integer exponents © math as a second language all rights reserved next #10 taking the fear...

Positive Integer Exponents

© Math As A Second Language All Rights Reserved

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#10

Taking the Fearout of Math

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In our discussion of the development of our number system we talked about how

even using place value notation it became cumbersome to denote large numbers.

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For example, an important constant in the study of chemistry is known as Avogadro's Number, which is the

number of atoms in a mole of water (a mole of water weighs a little bit more

than a half ounce). In place value notation it is approximately…

600,000,000,000,000,000,000,000.

So we invented a new notation for expressing powers of 10. Specifically, we

observed that every time we multiply a whole number by 10 we annex a 0

in the ones place.

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And since…

600,000,000,000,000,000,000,000 = 6 × 100,000,000,000,000,000,000,000

…we see that to obtain Avogadro’s Number in place value notation, we

multiply 6 by 23 factors of 10.

To abbreviate writing the product of23 factors of 10, we invented

the notation 1023.

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We referred to 10 as the base and to 23 as the exponent and, as we just saw, in place value notation it is a 1 followed by

23 zeroes. Hence, in exponential notation, Avogadro’s Number is written

as 6 × 1023.

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600,000,000,000,000,000,000,000.

Think for a moment about how you would answer the question…

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“How much is twenty 2’s?”

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Cautionary Note

Although “40” is the answer that most people would give (and it’s also

the answer that is usually accepted as being correct), the wording of

the question leaves much to be desired!

Most people tend to hear the question as if it were…

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“How much is the sum of twenty 2’s?”

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Cautionary Note

However, while it might not seem obvious at first, there are times when we are

interested in answering the question…

“How much is the product of twenty 2’s?”

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Cautionary Note

There is a huge difference between the product of a given number of 2’s and the

sum of the same number of 2's.

For example, the sum of twenty 2’s is 40, and granted that the computation is

tedious we can show that the product of twenty 2’s is 1,048,576.

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Definition

In the same way that we wrote 1020 to indicate the product of 20 ten’s, weuse the notation 220 to indicate the

product of 20 two’s.

If b is any number and n is any positive integer, we use the notation bn to denote

the product of n factors of b. We refer to b as the base and to n as the exponent and

to bn as the nth power of b.

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next So, for example, 43 means 4 × 4 × 4 or 64. In this case 4 is the base, 3 is the exponent and 64 is the 3rd power of 4.

Be careful not to confuse the 3rd power of 4 with the 4th power of 3. The two

numbers are not the same.

Notes

The 3rd power of 4 = 43 = 4 × 4 × 4 = 64

The 4th power of 3 = 34 = 3 × 3 × 3 × 3 = 81

while…

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Notice that as the exponent increases it becomes increasingly more cumbersome

to compute the power of the base.

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In essence, to compute the value of 44, we first have to know the value of 43; to compute the value of 45, we first have to know the value of 44; and to compute the

value of 46, we first have to know the value of 45.

Notes

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So, as you can see, if we use pencil and paper computation, it would be exceedingly

cumbersome and time-consuming to compute, say, 420.

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For that reason, it is helpful to have a scientific calculator - that is, a calculator that has

a key that looks like…

Notes

xy

xy

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In that case, we could quickly compute the value of 420 by the following sequence

of keystrokes.

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Step 1: Enter 4

Notes

4

xyStep 2: Enter xy

Step 3: Enter 20

Step 4: Enter =

20

1,099,511,627,776

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The calculator display tells us that 420 = 1,099,511,627,776

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Students might appreciate the power of the calculator by seeing how computing

the answer “long hand” quickly becomes tedious and time consuming (not to mention, the chances of making a

computational error during the process).

Notes

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However, the calculator is not a cure-all in the sense that while it can carry

out the instructions you give it, it cannot tell you whether the instructions

you gave it were correct.

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For example, suppose we flip a penny, a nickel and a dime, and that we want to know

the number of different ways in which the coins can turn up “heads” or “tails”.

As we shall see below, the correct answer is given by 2 × 2 × 2 and not by 2 + 2 + 2.

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More specifically, looking at only the penny, we realize that it can turn up either

“heads” or “tails”. There are just two possible outcomes. In terms of a chart…

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Penny

Outcome #1 heads

Outcome #2 tails

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Notice that the labels “Outcome 1” and “Outcome 2” are used simply

to count the possible outcomes.

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It doesn’t mean, for example, thatOutcome 1 is more likely or more

desirable than Outcome 2.

Note

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Regardless of which of the two outcomes occurs when the penny is tossed, the nickel can turn up in two possible ways, namely, either “heads” or “tails”. This doubles the

number of possible outcomes. That is, whichever of the two ways the penny turns

up, the nickel can be either “heads” or “tails”. Again, in terms of a chart…

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Outcome #1 heads heads

Outcome #2 tails heads

Penny Nickel Outcomes

Outcome #3 heads tails

Outcome #4 tails tails

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Continuing in this way, we see that the same process applies with respect to the dime. No matter which of the above

four possible outcomes occurs, the dime can turn up either “heads” or “tails”.

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This again doubles the number of possible outcomes.

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In terms of a chart…next

Penny Nickel Outcomes Dime

Outcome #1 heads heads heads

Outcome #2 tails heads heads

Outcome #3 heads tails heads

Outcome #4 tails tails heads

Outcome #5 heads heads tails

Outcome #6 tails heads tails

Outcome #7 heads tails tails

Outcome #8 tails tails tails

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In the above case, listing the possible outcomes and then counting them is

a simple process, but one that can quickly become tedious. Namely, with each

additional coin, we double the number of possible outcomes.

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In terms of actually listing the possible outcomes, it means that with each

additional coin, we would double the number of rows in our chart.

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In particular, if we want to find the number of outcomes we could obtain ifa fair coin was tossed 20 times (or if 20

coins were tossed once), we wouldhave to compute the value of 220.

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So while it is potentially “dangerous” touse the calculator for solving problems you don’t understand, the calculator is a great device for relieving you of the

need to do tedious computations.

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In other words the calculator will not tell you that the correct instruction is to compute 220 (as opposed to, say, 2 × 20),

but it will quickly show you that 220 = 1,048,576

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There is a big difference between knowing how to count and

knowing what to count..

Note

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In our previous discussion we saw that for a small number of flips of a coin, it was easy to count the number of possible

outcomes simply by listing them.

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However, while it might have been tedious to compute the product of twenty 2’s, it’s a

lot more tedious to list the 1,048,576 possible outcomes that can occur

if 20 coins are flipped!

Notes

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With respect to our present discussion, it wasn’t important from a mathematical

point of view that we were flipping coins.

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The more important thing is that we were dealing with an event that had only

two possible outcomes.

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The same mathematical reasoning can be applied to guessing answers on a true-false type of quiz (each statement must be either true or false but not both) to constructing

sets (each object can either belong to a set or not; one or the other but not both); to

betting on athletic events where there are no ties (that is, a team can either win or lose;

one or the other, but not both).

Notes

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To list the possible outcomes of guessing on a test that has three “true-false” questions, all we have to do is take the chart we made for

“flipping a coin 3 times” and (1) replace the words “heads” and “tails” by “true” and “false”,

respectively, (2) replace the “coin” in the column heading by “Question”…

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Question 1 Question 2 Outcomes Question 3Outcome #1 true true trueOutcome #2 false true trueOutcome #3 true false trueOutcome #4 false false trueOutcome #5 true true falseOutcome #6 false true falseOutcome #7 true false falseOutcome #8 false false false

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Because virtually everyone has experienced such events, we have so far

limited our discussion of exponential growth to flipping coins

and looking at true/false questions.

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However, exponential growth plays animportant role in many aspects of life,

including economics, science andfinancial planning.

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For our purposes at the elementary school level, it’s probably best to illustrate this with an application to money, which is something all students can relate to.

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So, for example, suppose you are 25 years old and that you want

to retire at age 60.

Investments

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You have been lucky enough to find an investment fund that will double the

amount of money you have in the account every 7 years.

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You have $10,000 that you can invest now, and you want to know how much

your investment will be worth when you retire (that is, 35 years from now).

Investments

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The chart below gives us the answer…

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Time Amount in Your Retirement Fund

Now $10,000

After 7 years $20,000





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Notice that the doubling process doesn’t depend on how much you have invested

in the fund.

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Since every 7 years your investmentdoubles and since 35 is the 5th multiple

of 7, in 35 years your investmentwill double five times. That is, your original investment is 25 or 32 times

what you have now.

Notes

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In the present illustration, the original investment happened to be $10,000; and in

this case 32 × $10,000 = $320,000.

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Notice the special property of exponential growth.

Notes

That is, every 7 years the investment becomes double what it was 7 years

earlier; not from what it was at the start of the original investment.

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After 7 years it is the $10,000 that doubles but after the 14th year it is the

$20,000 that doubles.

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Notice, then, that if at the end of the 35th year you decided to renew the

investment for another 7 years, it is the $320,000 that will double!

Notes

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Science and mathematics are in themselves, neither good nor bad. The principles are the same whether

they are applied to “good” things or to “bad” things.

nextEnrichment

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For example, the fact that 220 is greater than 1,000,000 tells us that if a person went out and helped 2 people, and then each of

these 2 people went out and helped 2 other people, etc.; by the 20th link in this

“chain” over one million people would have been helped with nobody having to

help more than 2 people!

Enrichment

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Yet the same mathematics tells us that if a person infected 2 people with AIDS;

and then each of these 2 people infected 2 other people with AIDS, etc.; by the 20th link in the chain over one million people would have been infected by

AIDS, with no one having to infect more than 2 people.

Enrichment

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Which way the mathematical result is used depends on society as a whole; not just on the scientist and mathematician.

This is why the humanities, the social sciences and the physical sciences

should be studied as a unified whole rather than in fragmented form.

Enrichment

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In the next presentation, we will begin

a discussion of the arithmetic of whole number exponents.


Arithmetic of Whole Number Exponents

positive integer exponents © math as a second language all rights reserved next #10 taking the fear...

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