Download Lesson 10 Keystone - Adjective Noun Math

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Keystone
Illustrations
Set 10
© 2007 Herbert I. Gross
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Instructions for the
Keystone Illustrations
You will soon be assigned problems to test
whether you have internalized the material
in Lesson 10 of our algebra course.
The Keystone Illustrations below are
prototypes of the problems you'll be doing.
© 2007 Herbert I. Gross
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Preface
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.
© 2007 Herbert I. Gross
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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.
Be sure that you understand how every
statement that's made in the proof is an
accepted property of the number system.
© 2007 Herbert I. Gross
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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”.
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.
© 2007 Herbert I. Gross
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You knew that since 3 – 3 = 0,
x + 3 – 3 = x, and in vertical form your
solution might then look like…
x + 3 = 7
– 3 –3
x
= 4
Probably no one would have disagreed
with your proof.
© 2007 Herbert I. Gross
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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).
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.
© 2007 Herbert I. Gross
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To this end, suppose someone who
accepted our rules of the game wanted us
to prove that if x + 3 = 7, then x = 4.
-- 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) + -3 = 7 + -3 (= 4).
-- Then by the associative property for
addition we are allowed to rewrite…
(x + 3) + -3 as x + (3 + -3).
-- Hence, we may rewrite (x + 3) + -3 = 4 as…
x + (3 + -3) = 4.
© 2007 Herbert I. Gross
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-- by the additive inverse we know that
3 + -3 = 0. Hence, by substitution we may
replace x + (3 + -3) by x + 0.
-- Therefore, we may rewrite x + (3 + -3) = 4
in the form x + 0 = 4.
-- 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 = 4.
© 2007 Herbert I. Gross
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More formally To prove that if x + 3 = 7
then x = 4.
Proof
Statement
Reason
(1) (x + -3) + 3 = 7 + -3 (=4)
(1) Substituting 7 for x + 3
(2) (x+ 3) + 3 = x + (3 + -3)
(2) Associative Property (+)
(3) x + (3 + -3) = 4
(3) Substituting (3) into (1)
(4) 3 + -3 = 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)
© 2007 Herbert I. Gross
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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.
Caution
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…
© 2007 Herbert I. Gross
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Keystone Illustrations for Lesson 10
#2 We have all been taught that 3 + 2 = 5
However, suppose a skeptic were to ask, “What
rule tells us that 3 + 2 = 5?” To answer this, we
could start with the fact that since 1 is a
number, the closure property for addition tells
us that 1 + 1 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.
© 2007 Herbert I. Gross
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Continuing in this way we have, by
definition… 1 + 1 = 2; 2 + 1 = 3; 3 + 1 = 4;
4 + 1 = 5; 5 + 1 = 6; 6 +1 = 7; 7 + 1 = 8; etc.
Hence, if we start with the expression…
3+2
we may replace 2 above by 1 + 1 to obtain…
3 + 2 = 3 + (1 + 1)
By the associative property for addition
we know that…
3 + (1 + 1) = (3 + 1) + 1
© 2007 Herbert I. Gross
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Therefore by substitution we may replace
3 + (1 + 1) by its value (3 + 1) + 1 to obtain…
3 + 2 = (3 + 1) + 1
By definition…
3+1=4
So we may replace 3 + 1 by its value 4 to
obtain…
3+2=4+1
And since by definition 4 + 1 = 5, we
may rewrite 3 + 2 = 4 + 1 as…
3+2=5
© 2007 Herbert I. Gross
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More formally To prove that 3 + 2 = 5
Proof
Statement
Reason
(1) 1 is a number
(1) Multiplicative Identity
(2) 1 + 1 is a number
(2) Closure for Addition
(3) 1 + 1 = 2
(3) Definition of 2
(4) 3 + 2 = 3 + (1 + 1)
(4) Substituting 1+1 for 2
(5) 3 + (1 + 1) = (3 + 1) + 1
(5) Associative Property (+)
(6) 3 + 2 = (3 + 1) + 1
(6) Substituting (5) into (4)
(7) 3 + 1 = 4
(7) Definition of 4
(8) 3 + 2 = 4 + 1
(8) Substituting (7) into (6)
(9) 4 + 1 = 5
(9) Definition of 5
(10) 3 + 2 = 5
(10) Substituting (9) into (8)
© 2007 Herbert I. Gross
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Note
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 3 + 2 in the form…
© 2007 Herbert I. Gross
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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… And finally we may move the
remaining tally mark on the right closer to
the left to obtain…
5
43
© 2007 Herbert I. Gross
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2
1
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Preface to the Next
Illustration
Consider the equality…
3 dimes + 2 nickels = 40 cents.
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…
3 + 2 = 40;
which would seem to be nonsense!
© 2007 Herbert I. Gross
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Based on the ambiguity as to whether
3 + 2 = 5 or 3 + 2 = 40, it seems that the
equality 3 + 2 = 5 should be emended to
read…
3 + 2 = 5 provided that 3, 2, and 5 are
adjectives modifying the same noun.
If we now use x as a generic name for
the noun, the above equality becomes…
3x + 2x = 5x
© 2007 Herbert I. Gross
Therefore…
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Keystone Illustrations for Lesson 10
#3 In terms of our rules of the game, we
now know that 3 + 2 = 5
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
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To prove that if 3 + 2 = 5
then 3x + 2x = 5x.
Proof
Statement
Reason
(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) 3 + 2 = 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)
© 2007 Herbert I. Gross
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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
0 + 0 + 0, which is clearly 0.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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A Proof That a0 = 0
Since 0 is the additive identity , we know
that…
0+0=0
Hence by substitution, we may interchange
0 + 0 and 0 in any mathematical expression.
In particular, we may multiply both sides
of the above equation by a to obtain…
a(0 + 0) = a0
By the distributive property we know that…
a(0 + 0) = a0 + a0
© 2007 Herbert I. Gross
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Hence, again by substitution, we may
replace a(0 + 0 ) by a0 + a0 to obtain…
a(0
a0 ++ a0
0) = a0
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…
a0 = b
© 2007 Herbert I. Gross
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Substituting b for a0 we obtain…
a0b++a0
b = a0
b
If we now subtract b from both sides of
the equation, we see that…
b + b= b
–b
= –b
b = 0
© 2007 Herbert I. Gross
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And since b = a0, and b = 0 we see
that…
a0
b = a0
0
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…
© 2007 Herbert I. Gross
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To prove that a0 = 0
Proof
Statement
Reason
(1) 0 + 0 = 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
© 2007 Herbert I. Gross
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A Possible
Oversight
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.
In our more formal proof, steps (6) through
(9) rectified this oversight.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross