rates to introduce signed numbers © math as a second language all rights reserved next #9 taking...

Rates to Introduce

Signed Numbers

© Math As A Second Language All Rights Reserved

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#9

Taking the Fearout of Math

-3

+3

Does it seem strange that the ancient Greeks were able to do extensive work

with rational numbers (fractions) but did not know that negative numbers existed?

The reason is that they viewed numbers as being lengths and no length could be

“less than nothing”. That is, how could apiece of string be so short that even if it

was 2 inches longer it would have “zero length”?

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In text books, signed numbers are often introduced in terms of the number line (a special case of which is a liquidthermometer) or the business model

(namely profit/loss).

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Specifically, a $3 loss is referred to as negative 3 while a $3 profit is referred to

as positive 31.note

1 However the phrase “a $3 loss” takes away the need to use the word “negative”. In a similar way, saying “3 degrees below zero” takes away the need to talk

about “negative 3 degrees”.

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In our earlier lessons we stressed the concept of rates. We may also apply the

concept of signed numbers to rates. Thisis illustrated in the following problem.

In a certain town the rate of change of the population is 5,000 persons per year.

If the population of the town this year is55,000 people, what was the population of

the town last year?

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The answer is either 50,000 or 60,000. We are not given enough information to

arrive at a unique answer. Namely, we have to know whether the change in population

represented an increase or whether it represented a decrease. If the population is

decreasing at a rate of 5,000 persons per year, then a year ago the population would

have been 60,000. However if the population is increasing at a rate of 5,000

persons per year, then a year ago the population would have been 50,000.

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Thus, in terms of our study of rates, another way to visualize signed numbers is

by having positive represent an increase and negative a decrease. 0 would represent

the fact that no change took place.

This gives us a “real world” reason as to why +3 and -3 are connected by the fact

that +3 + -3 = 0. For example, if youmake a $3 profit on one transaction (+3) and then you incur a loss of $3 on the

second transaction (-3), the net result is that you “broke even” (0).

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Recall that in our treatment of rational numbers, we first established a need for

“inventing” them (for example, to be able to perform the division 5 ÷ 3), after which we

developed the rules of the game thatgoverned the arithmetic of rational numbers.

To apply a similar discussion to the invention of signed numbers, let’s look at

the “fill-in-the-blank” problem…

3 +___ = 0

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As of now in our course, 0 is the smallest number, and since 3 + 0 = 3,

it means that if we add any number to 3 the answer will be greater than 3.

So we can either say that…

There is no reason for us to have to worry about this since no number can be less than 0. So let’s just leave the number

system as is.

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or…

If we have made a $3 profit on one transaction and then a $3 loss on another transaction, we have “broken even” (that

is, we made a $0 profit). In other words…

$3 profit + $3 loss = $0 profit (or loss).

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Therefore, if we want a mathematical model that is applicable to the real world,

we have to invent a number system in which a fill-in-the-blank problem such as

3 +___ = 0 has a numerical answer. That is why the number -3 was invented.

We generalize this result by adding the property…

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If a is any number, there exists another number b, such that a + b = 0

In this case we refer to a and b as the opposites of one another (or, more formally,

the additive inverses of one another).

If a and b are opposites of one another, we often rewrite b as -a and/or a as -b.

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Because 0 + 0 = 0, we see that 0 is its own opposite. So while 0 is neither positive nor negative it does have an opposite

(namely, itself).

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It is not important which we call positive and which we call negative. What is

important is that one is the “opposite” of theother. For example, suppose you go to a candy store and buy a $3 box of candy.

When you leave the store you have 1 morebox of candy but $3 less than when you came into the store. On the other hand,

from the shop keeper’s perspective, whenyou leave the store he has 1 less box of

candy but $3 more than when you came in.

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Notes

Notice that the opposite of a number can be positive.

For example, the opposite of -3 is +3.

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Notes

It is often customary to omit the positive sign when we talk about signed numbers. Namely, when we talk about buying, say, 3

apples, we mean 3 more than 0 (that is, 3 more than 0 apples)

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Many textbooks use the notation (-3) rather than -3. The parentheses are used to indicate that the sign is associated with the number

rather than with the operation of subtraction. Using parentheses is cumbersome.

Additionally we prefer the notation -1 to emphasize the difference between subtraction, opposite and the sign of a number. By writing

the sign as a “superscript” we are trying to indicate without the use of parentheses that

the sign is associated with the number.

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Notes

We do not read -1 as “minus 1”. Rather we read it as “negative 1”. In other words, when we say “minus” it means that we are going

to perform subtraction (as in 5 – 3 being read as “5 minus 3”). On the other hand, we

say “negative 3” when we are referring to the sign of a signed number, and we say the opposite of -3 when we are referring

to changing the sign of -3.

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Notes

For example, we would read -3 – -b as “negative 3 minus the opposite of b”.

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To view signed numbers in terms of the “real world”, we often use at least one of the

following models…

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In Summary

The “Business Model”

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The Temperature Model

The Directed Distance Model

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The “Business Model”next

In terms of this model 3 +___ = 0 would be the fill in the blank form for answering

a question such as “If the first transaction resulted in a

$3 profit, what must the second transaction have to be in order for the net result of these two transactions to

be a $0 profit?”

+3-3

Profit/Loss

In terms of this model 3 +___ = 0 would be the fill in the blank form for answering a question such as,

“If the temperature increases by 3°C in the first hour, what must it

do during the second hour in order for the temperature to return

to what it was originally?”

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The Temperature Model

0+1+2+3+4+5+6+7

-1-2-3-4-5

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In terms of this model 3 +___ = 0 would be the fill in the blank form for answering

a question such as, “If you move 3 units to the right of where

you were, what must you do to returnto your starting point?”

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The Directed Distance Modelnext

By convention “to the right of 0” is positive and “to the left of 0” is negative.

0 3

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In general, all of the models are basically rate models (which we may call

“increasing/decreasing” models) in which“increasing” means that the sign is

positive and “decreasing”means that the sign is negative.

In these cases, 0 is used to mean that there was no change.

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In the next presentation, we will begin a

discussion of how we add signed numbers.


Adding Signed Numbers

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