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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 3
Introduction to Signed Numbers: Defining and Adding Signed Numbers Introduction to Signed Numbers: Defining and Adding Signed Numbers + - © 2007 Herbert I. Gross next
Does it seem strange that the ancient Greeks were able to do extensive work with rational numbers (fractions), but didn’t know that negative numbers existed? © 2007 Herbert I. Gross next The reason is that they viewed numbers as being lengths. That a length could be “less than nothing” made no sense to them. How could a piece of string have less than “zero length”? next
In textbooks, signed numbers are often introduced in terms of the number line (a special case of which is the old fashioned mercury thermometer) or the business model (namely profit/loss). © 2007 Herbert I. Gross next More specifically, a $3 loss is referred to as negative 3 while a $3 profit is referred to as positive 3. 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”.
In our earlier lessons, we stressed the concept of rates. We can also apply the concept of signed numbers to rates. This is illustrated in the following problem. © 2007 Herbert I. Gross next 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 is 55,000 people, what was the population of the town last year? next
© 2007 Herbert I. Gross next The answer is either 50,000 or 60,000. More specifically, we were not given enough information to arrive at a unique answer. We were not told whether the change in population represented an increase or a decrease.
© 2007 Herbert I. Gross next 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.
Thus, another way to visualize signed numbers is by having positive represent an increase and negative a decrease. 0 represents the fact that no change took place. © 2007 Herbert I. Gross next Note
However, mathematicians require definitions that do not depend on the model that is being used. © 2007 Herbert I. Gross next Thus, they have special definitions and rules for everything having to do with signed numbers. So numbers (for example, 3) are represented as + 3 when they are positive and as־3 when they are negative.
Although mathematicians require rules, they often use physical models to insure that the rules are “realistic”. © 2007 Herbert I. Gross next The mathematical connection between… + 3 and ־3 is that ־3 = 0. For example… next
A $3 profit followed by a $3 loss results in no net change. © 2007 Herbert I. Gross next That this rule is “realistic” can be verified by such observations as… A 3 degree increase in temperature followed by a 3 degree decrease in temperature results in no net change in the temperature.
We refer to + 3 and־3 as being opposites (but more formally, the additive inverses) of one another). In more general terms… © 2007 Herbert I. Gross next And because = 0, we see that 0 is its own opposite. So while 0 is neither positive nor negative, it does have an opposite (namely itself). If a and b are any two numbers, they are called opposites of one another if a + b = 0. In such a case: we can refer to b as – a and to a as – b.
Notice that the opposite of a number can be positive. © 2007 Herbert I. Gross next Caution For example… The opposite of־3 is + 3. However, we frequently omit the positive sign when we write signed numbers. For example, when we talk about buying 3 apples, everyone understands we are buying 3 more than 0 apples. next
Rather than writing־3, many text books use the notation (–3). The parentheses are used to indicate that the sign is associated with the number rather than with the operation of subtraction. © 2007 Herbert I. Gross next Notation
However, in our course we will use the notation ־ 3 because we want to emphasize the differences between subtraction, opposite, and the sign of a number. © 2007 Herbert I. Gross next Notation Writing the sign as a “superscript” also emphasizes that the sign is part of this particular number. next
Thus, we do not read־3 as “minus 3”. Rather, we read it as “negative 3”. © 2007 Herbert I. Gross next Note 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”). next
© 2007 Herbert I. Gross next Note We say “negative 3” when we are referring to the sign of a signed number. In summary… we would read “ ־3 – ־b” as… “negative 3 minus the opposite of b”. And we talk about the opposite of 3 when we are referring to changing the sign of + 3. next
Interestingly, which number we call positive and which we call negative is not important. © 2007 Herbert I. Gross next Note For example, suppose you go to a candy store and buy a $3 box of candy. When you leave the store, you have 1 more box of candy but $3 less than when you came into the store. next The important point is that one is the opposite of the other. On the other hand, from the shop-keeper’s point of view, he has 1 less box of candy but $3 more than when you came in. next
While the definition of b – c as the number we must add to c to obtain b as the sum is an improvement of the “take away” concept, it brings with it a new challenge. © 2007 Herbert I. Gross next 2 – 3 = ? or equivalently… 3 + ? = 2 From the ancient Greeks’ point of view, the least number was 0, and since = 3, they saw no number that could be added to 3 to get 2 as the sum. next For example, consider a question such as…
Their math looked at the 2 – 3 = ? question from the view that it is not possible to take 3 pieces of candy from a dish that contains only 2 pieces of candy. © 2007 Herbert I. Gross next On the other hand, we have different ways of looking at the 2 – 3 = ? question. For example, temperature can increase by 2° one hour, and then decrease by 3° the next hour. Thus, the net result after the 2 hour period is a temperature decrease of 1°.
If we start the day at some temperature, and the temperature then increases by 2 degrees; and then, decreases by 3 degrees; the temperature ends at 1 degree less than the temperature at which we started. © 2007 Herbert I. Gross next starting point ending point next Thermometer Illustration
To view signed numbers in terms of the “real world”, we often use at least one of the following models. © 2007 Herbert I. Gross next There is the “business model" (wherein “profit” is positive and “loss” is negative). There is the “temperature model” (wherein “above zero” is positive and “below zero” is negative).
© 2007 Herbert I. Gross next There is also the “directed distance model” (wherein “to the right of 0” is positive and “to the left of 0” is negative). The unifying thread for all of these is the rate model (which we may call the increasing/decreasing model) wherein “increasing” means that the sign is positive and “decreasing” means that the sign is negative, and 0 means that there was no change. 0 positivenegative
Our adjective/noun theme (introduced in the prelude to this lesson) makes it easy for us to add signed numbers, provided the numbers all have the same sign. © 2007 Herbert I. Gross next ־ 3 + ־ 2 = ־ 5 because the numbers, 3 and 2, are both modifying “negative”. Example 3 negative + 2 negative 3 less than less than 0 or next negative less than 0 5 5
When we write - 4, we understand that 4 is the adjective and the sign is the noun. © 2007 Herbert I. Gross next Note However, there are other exceptions. This is an exception to English grammar, wherein the adjective usually comes before the noun. For example, when we abbreviate “4 dollars” we write it as “$4”, not as “4$” (although we could have written it that way). next
In summary, a signed number such as ־ 4 is phrase consisting of an adjective (in this case, 4) and a noun (in this case, negative). © 2007 Herbert I. Gross next Note Thus, when we deal with signed numbers there are only two nouns… positive and negative. next
We often refer to the adjective as either the absolute value of the signed number, or as the magnitude of the signed number. © 2007 Herbert I. Gross next Vocabulary/Notation Thus, the absolute value of both + 3 and־3 is 3, and we use a special symbol to denote the absolute value, namely, | + 3| or |־3|. In summary… |+3| = |־3| = 3 next
There are several ways to visualize why ־ 3 + ־ 2 = ־ 5. © 2007 Herbert I. Gross next -- A $3 loss followed by a $2 loss is a $5 loss. -- If the temperature decreases 3°F the first hour and then it decreases 2°F the second hour, the net change in the temperature after the two hours is a decrease of 5°F. Examples
Other ways to visualize why ־ 3 + ־ 2 = ־ 5. © 2007 Herbert I. Gross next -- If we move 3 units from right to left; and then move another 2 units from right to left; we have moved a total of 5 units from right to left. -- A price decrease of $3 per item, followed by another price decrease of $2 per item, is a total price decrease of $5 per item.
These models make it relatively easy to visualize why the rules for adding signed numbers follow the same rules that apply to whole numbers. © 2007 Herbert I. Gross next
© 2007 Herbert I. Gross next That is: if the first transaction results in a $4 loss and the second transaction results in a $5 loss, the net result is a $9 loss; which is the same net result we’d have if the first transaction had been the $5 loss, and the second transaction had been the $4 loss. For example, with respect to the commutative property of addition: the net result of two business transactions does not depend on the order in which we view the two transactions.
To summarize the results of these examples in a way that is independent of the model that was used, mathematicians adopt the following rule… © 2007 Herbert I. Gross next To add two numbers that have the same sign, 3 negative + 2 negative 5 negative and keep the common sign (the noun). we add the adjectives (magnitudes, absolute values)
However, a problem that we encounter is that we have not yet been given a rule that tells us how to add two numbers if they have different signs. For example, if we use the adjective/noun theme to compute the sum of, say, ־ 3 and + 2, it would look like. © 2007 Herbert I. Gross next 3 negative + 2 positive And in this case we cannot add 3 and 2 because they are modifying different nouns.
By revisiting the additive inverse property with respect to the previous illustrations, we see for example that… © 2007 Herbert I. Gross next -- A $3 profit followed by a $3 loss results in our “breaking even”. -- A price increase of $3 per item followed by a price decrease of $3 per item results in the item maintaining its original price.
© 2007 Herbert I. Gross next -- If the temperature increases 3°F the first hour, and then it decreases 3°F the second hour, then there is no net change in temperature after the two hours. -- If we move 3 units from left to right on the number line, and then move another 3 units from right to left, we have returned to the point from which we started.
The fact that a number plus its opposite is zero makes it fairly simple to add signed numbers even when they don’t all have the same sign. © 2007 Herbert I. Gross next Suppose we want to form the sum If we use our profit/loss model we have that… Example ProfitLoss First Transaction Second Transaction Net next = ? $8$8 $5$5
We know that if the situation had been a $5 loss and a $5 profit, the net would have been $0. This is simply the profit/loss model for saying that = 0. That is… © 2007 Herbert I. Gross next ProfitLoss First Transaction Second Transaction Net next ־ = 0 $5$5 $5$5 $0
With this as a hint, we simply view the $8 loss as the sum of a $3 loss and a $5 loss. © 2007 Herbert I. Gross next ־ = - 3 ProfitLoss First Transaction$8$8 Net Second Transaction$5$5 $3$3 $5$5 $3$3 next The $5 profit and the $5 loss then “cancel” one another, and we see that the net result is a $3 loss.
Leaving aside the profit/loss model and using only our known properties of arithmetic, the computation looks like the following… © 2007 Herbert I. Gross next = ( ) = ( ) = = = - 3 next
In going from… © 2007 Herbert I. Gross next ( ) = ( ) = next to… we assumed that the associative property for addition (regrouping) could be extended to signed numbers. That this is plausible can be seen from the fact that the net result of a $3 loss, a $5 loss, and a $5 profit is a $3 loss no matter in which order the transactions occurred. Note
To summarize the above process in a way that is independent of any model, mathematicians use the following definition… © 2007 Herbert I. Gross next Key Point To add two signed numbers that have different signs, we subtract the lesser magnitude (adjective) from the greater one (in this case, we obtain 8 – 5 = 3), and the answer, - 3, keeps the sign of the number that had the greater magnitude (in this case - 8). next
Notice that we never subtract a greater magnitude from a lesser magnitude. © 2007 Herbert I. Gross next Key Note That is: the sign of the answer, whether positive or negative, comes from the sign of the number that has the greater adjective. next
Enrichment Note © 2007 Herbert I. Gross Imagine that you have colored chips, one color to represent positive, and the other color to represent negative. To add, we just “amalgamate” all chips that have the same color; and the additive inverse property allows us to remove any pair of chips that have opposite colors. PN Chip Model
So suppose we use P to denote a “positive” chip and N to denote a negative chip. In this way would look like… next © 2007 Herbert I. Gross NNNNNNNNNNNN …and we see that here we have 12 “negative” chips; thus indicating that… = next
On the other hand, if we had , the chip model would look like… © 2007 Herbert I. Gross PPPPPNNNNNNN and if we now cancel “opposite pairs”, next we see that… = - 2.
The different models for addition of signed numbers give us ways to visualize such statements as… © 2007 Herbert I. Gross next = + 8 next = = = - 2
The profit and loss model shows a $5 ( + 5) profit and a $3 ( + 3) profit would net an $8 ( + 8) profit. © 2007 Herbert I. Gross next = + 8 ProfitLoss First Transaction Second Transaction $5$5 $3$3 Net$8$8
If we start at 0 and move 5 units to the left of zero ( - 5) on the number line and then move 3 more units to the left ( - 3), we end at - 8. © 2007 Herbert I. Gross next = next
The temperature model shows that if the temperature increases by 5 ( + 5) degrees and then decreases by 3 ( - 3) degrees: the net change would be an increase of 2 ( + 2) degrees. © 2007 Herbert I. Gross next = + 2 next
Using chips, 5 negative chips ( - 5) plus 3 positive chips ( + 3) chips would leave 2 negative chips ( - 2). © 2007 Herbert I. Gross next = - 2 NNNNNPPP
A signed number is a number that has both a size and a direction. © 2007 Herbert I. Gross next Summary The size is referred to as either the magnitude or as the absolute value; and the direction (which is either positive or negative) is referred to as the sign. next
In terms of our adjective/noun theme, the size of the signed number is the adjective, and the sign of the number is the noun. For example, + 8 denotes an increase of 8 while - 8 represents a decrease of 8. © 2007 Herbert I. Gross next So to add two signed numbers that have the same sign (that is: they modify the same noun), we add the two magnitudes and keep the common sign. For example … = + ( ) = + 13, and = - ( ) = - 13
If the two numbers have different signs, we subtract the lesser magnitude from the greater magnitude and we keep the sign of the number that had the greater magnitude. © 2007 Herbert I. Gross next Thus, for example: when we add - 8 and + 5, the size will be obtained by subtracting the lesser number (5) from the greater number (8). We know that the sign of the sum will be negative because 8 is greater than 5. In other words… = - ( 8 – 5) = - 3
© 2007 Herbert I. Gross next Care is needed not to confuse adding two signed numbers that have opposite signs with subtracting two signed numbers. To repeat… When we add two numbers that have different signs, we subtract the lesser magnitude from the greater magnitude. (We will discuss subtraction of signed numbers in our next lesson.)
© 2007 Herbert I. Gross next $8 Profit $5 Loss $3 Profit next In other words, the sum a + b is less than a if b is negative. In terms of profit and loss = + (8 – 5) = + 3 represents the fact that the net result of an $8 profit followed by a $5 loss results in only a $3 profit.
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Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross By Herbert I. Gross and Richard A. Medeiros next Set 9.
The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard.
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