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Keystone Illustrations Keystone Illustrations next Set 10 © 2007 Herbert I. Gross
You will soon be assigned problems to test whether you have internalized the material in Lesson 10 of our algebra course. Instructions for the Keystone Illustrations next © 2007 Herbert I. Gross The Keystone Illustrations below are prototypes of the problems you'll be doing. next
Most likely the concept of mathematics as a game is new to you, and consequently it may take time for you to become comfortable with how this game is played. Therefore, rather than a keystone problem, we will give several different types of illustrations in this presentation to show what we mean by a proof. Preface next © 2007 Herbert I. Gross
In many ways, it is beyond the scope of an introductory algebra course to spend too much time on proofs. Yet it is important to understand the nature of a proof and how it uses “facts" to derive other “facts”. We will limit our illustrations to showing how our rules justify some of the things that we ordinarily take for granted in an algebra course. next © 2007 Herbert I. Gross Be sure that you understand how every statement that's made in the proof is an accepted property of the number system. next
Once we feel that the idea has been adequately presented; we will become more informal and resort to proofs only when we feel that a statement is not “self-evident”. © 2007 Herbert I. Gross Keystone Illustrations for Lesson 10 #1 Let's say you were called upon to find the value of x for which x + 3 = 7. Quite likely you realized that by subtracting 3 from both sides of the equation you would obtain the result that x = 4. next
You knew that since 3 – 3 = 0, x + 3 – 3 = x, and in vertical form your solution might then look like… © 2007 Herbert I. Gross x + 3 = 7 next – 3 – 3 x = 4 Probably no one would have disagreed with your proof.
However suppose someone who accepted our rules of the game had observed that by the closure property of addition x – 3 (i.e., x + - 3) is one number. Hence, when you subtract 3 from it, the correct way to indicate this is as (x – 3) + 3 rather than as x + (3 – 3). next © 2007 Herbert I. Gross next So no matter how obvious the proof may seem to you, the obligation is to show the skeptic that your approach was justified by the accepted rules of the game.
To this end, suppose someone who accepted our rules of the game wanted us to prove that if x + 3 = 7, then x = 4. © 2007 Herbert I. Gross next -- Starting with x + 3 = 7, we might begin by adding - 3 to x + 3. Then by replacing x + 3 by 7 (substitution), we could write that… (x + 3) = (= 4). -- Then by the associative property for addition we are allowed to rewrite… (x + 3) as x + ( ). -- Hence, we may rewrite (x + 3) = 4 as… x + ( ) = 4.
© 2007 Herbert I. Gross next -- by the additive inverse we know that = 0. Hence, by substitution we may replace x + ( ) by x Knowing that 0 is the additive identity tells us that x + 0 = x. Hence, we may replace the equation x + 0 = 4 by the equation x = Therefore, we may rewrite x + ( ) = 4 in the form x + 0 = 4.
© 2007 Herbert I. Gross To prove that if x + 3 = 7 then x = 4. next More formally Proof StatementReason (1) (x + - 3) + 3 = (=4)(1) Substituting 7 for x + 3 (2) (x+ 3) + 3 = x + ( )(2) Associative Property (+) (3) x + ( ) = 4(3) Substituting (3) into (1) (4) = 0(4) Additive Inverse Property (5) x + 0 = 4(5) Substituting (4) into (3) (6) x + 0 = x(6) 0 is the Additive Identity (7) x = 4(7) Substituting (6) into (5)
So what we've shown is that replacing (x + 3) – 3 by x is an inescapable consequence of the rules of the game. Therefore, we may now use it as a “fact” whenever we wish without having to demonstrate the validity of this again. next © 2007 Herbert I. Gross If we restrict our assumed knowledge to the accepted rules of the game, only the numbers 0 and 1 exist. In particular, the numbers 7, 3, and 4 do not yet exist. This leads us to our next illustration… next Caution
next © 2007 Herbert I. Gross Keystone Illustrations for Lesson 10 #2 We have all been taught that = 5 next However, suppose a skeptic were to ask, “What rule tells us that = 5?” To answer this, we could start with the fact that since 1 is a number, the closure property for addition tells us that is also a number. We call that number 2. Since 2 and 1 are numbers, so is 2 + 1; which we will call 3. Aside: By the additive inverse property once 1, 2, and 3 are numbers, so also are - 1, - 2, and - 3.
next © 2007 Herbert I. Gross Continuing in this way we have, by definition… = 5; = 6; 6 +1 = 7; = 8; etc. Hence, if we start with the expression… we may replace 2 above by to obtain… = 3 + (1 + 1) By the associative property for addition we know that… 3 + (1 + 1) = (3 + 1) + 1 next = 2; = 3; = 4;
next © 2007 Herbert I. Gross Therefore by substitution we may replace 3 + (1 + 1) by its value (3 + 1) + 1 to obtain… = (3 + 1) + 1 By definition… = 4 So we may replace by its value 4 to obtain… = And since by definition = 5, we may rewrite = as… = 5 next
© 2007 Herbert I. Gross To prove that = 5 next More formally Proof StatementReason (1) 1 is a number(1) Multiplicative Identity (2) is a number(2) Closure for Addition (3) = 2(3) Definition of 2 (4) = 3 + (1 + 1)(4) Substituting 1+1 for 2 (5) 3 + (1 + 1) = (3 + 1) + 1(5) Associative Property (+) (6) = (3 + 1) + 1(6) Substituting (5) into (4) (7) = 4(7) Definition of 4 (8) = 4 + 1(8) Substituting (7) into (6) (9) = 5(9) Definition of 5 (10) = 5(10) Substituting (9) into (8)
While our discussion in the above situation might seem a bit abstract, it captures the way young children tend to use their fingers to do addition. More specifically, if we use tally marks to represent fingers, we may view in the form… next © 2007 Herbert I. Gross Note next
This array can be regrouped by moving one of the two tally marks on the right closer to the three tally marks on the left to obtain… © 2007 Herbert I. Gross next And finally we may move the remaining tally mark on the right closer to the left to obtain…
Consider the equality… 3 dimes + 2 nickels = 40 cents. Preface to the Next Illustration next © 2007 Herbert I. Gross In this case 3, 2, and 40 are adjectives modifying the nouns dimes, nickels, and cents respectively. If we were to omit the nouns, the above equality would become… = 40; which would seem to be nonsense! next
© 2007 Herbert I. Gross Based on the ambiguity as to whether = 5 or = 40, it seems that the equality = 5 should be emended to read… = 5 provided that 3, 2, and 5 are adjectives modifying the same noun. next If we now use x as a generic name for the noun, the above equality becomes… 3x + 2x = 5x Therefore… next
© 2007 Herbert I. Gross Keystone Illustrations for Lesson 10 #3 In terms of our rules of the game, we now know that = 5 next We will now show that as a consequence 3x + 2x = 5x, where x is any number. In the spirit of deductive reasoning, one form of our proof would be…
© 2007 Herbert I. Gross To prove that if = 5 then 3x + 2x = 5x. next Proof StatementReason (1) 3x = x3 and 2x = x2(1) Commutative Property (×) (2) 3x + 2x = x3 + x2(2) Substitution (3) x(3 + 2) = x3 + x2(3) Distributive Property (4) 3x + 2x = x(3 + 2)(4) Substituting (3) into (2) (5) = 5(5) Previously Proved (6) 3x + 2x = x5(6) Substituting (5) into (4) (7) x5 = 5x (7) Commutative Property (×) (8) 3x + 2x = 5x(8) Substituting (7) into (6)
next © 2007 Herbert I. Gross Enrichment Illustration for Lesson 10 #4 In terms of working with 0 and 1, all we know from the listed properties is that a + 0 = a and a × 1 = a. Nowhere do we have a rule or a definition that tells us that for any number a, a × 0 =0. Of course, this “fact” might seem obvious to us from what we have learned in arithmetic. For example, we defined 3 × 0 to mean , which is clearly 0. next
© 2007 Herbert I. Gross However, there are many students in arithmetic who feel that a × 0 should be a. The reason lies in the “excuse" that is often given as to why a + 0 = a. Namely… “Since we didn't add anything to a the sum is still a. So why shouldn't the answer still be a if we multiply it by nothing?” However, the spirit of our game requires that we can only use results that are contained in the definitions and the rules we accepted; and a × 0 = 0 is not one of the rules we have accepted. next
© 2007 Herbert I. Gross If we are able to show that a × 0 = 0 follows inescapably from the rules of our game, it means that every student who has accepted the “rules”, as they were presented in this lesson, must accept as a fact that a × 0 = 0. We will prove below that a × 0 = 0, but for the sake of brevity we will write a × 0 in the form a0 next
© 2007 Herbert I. Gross Since 0 is the additive identity, we know that… A Proof That a0 = 0 Hence by substitution, we may interchange and 0 in any mathematical expression. In particular, we may multiply both sides of the above equation by a to obtain… next = 0 a(0 + 0) = a0 By the distributive property we know that… a(0 + 0) = a0 + a0
next © 2007 Herbert I. Gross Hence, again by substitution, we may replace a(0 + 0 ) by a0 + a0 to obtain… Since a and 0 are numbers, and since the numbers are closed with respect to multiplication, we know that a0 is a number. So, as an abbreviation, let b stand for the number a0. That is… next a0 + a0 = a0 a0 = b a(0 + 0) = a0
next © 2007 Herbert I. Gross Substituting b for a0 we obtain… If we now subtract b from both sides of the equation, we see that… next b + b = b a0 + a0 = a0 –b = –b b = 0
next © 2007 Herbert I. Gross And since b = a0, and b = 0 we see that… Thus, a0 = 0 is an inescapable conclusion based on the accepted properties of our number system. Therefore, it may be treated as a fact in our “game”. Summarized more formally… next a0 = 0b = a0
next © 2007 Herbert I. Gross To prove that a0 = 0 next Proof StatementReason (1) = 0(1) Additive Identity Property (2) a(0 + 0) = a(0)(2) Substitution (3) a(0 + 0) = a0 + a0(3) Distributive Property (4) a0 + a0 = a0(4) Substituting (3) into (2) (5) b + b = b(5) Definition of b (b = a0) (6) (b + b) + - b = b + - b(6) Substituting (5) into (b+b) + - b (7) b + (b + - b) = b + - b(7) Associative Property (+) (8) b + 0 = 0(8) Additive Inverse Property (9) b = 0(9) Additive identity Property (10) a0 = 0(10) Substituting a0 for b
In our informal discussion for proving that a0 = 0; when we arrived at the equation b + b = b, we nonchalantly subtracted b from both sides to obtain b = 0. However, none of our rules justified this assertion. next © 2007 Herbert I. Gross A Possible Oversight next In our more formal proof, steps (6) through (9) rectified this oversight.
© 2007 Herbert I. Gross Hopefully, the above situations help you internalize what we mean by a proof and why it's necessary that we have proofs. However, having done this, our strategy in this course will be to prove only those facts that might not seem to be obvious to us from our past experiences in mathematics. In other words, we shall continue to state as facts such things as: we may “subtract equals from equals” or “when adding numbers, the sum does not depend on how the terms are rearranged and/or regrouped” etc. More practice is left for the Exercise Set.
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