Student Learning Opportunities
Theorem 1.2 An angle inscribed in a semicircle is a right angle
1.3.4 The Finality of Proof
In mathematics when one produces a correct proof, one never has to question its truth again.
That is what mathematical proof is all about. It is about finality and knowing for sure that, for eternity, something is true or not true. That is very different from proof in the sciences, where, for the most part, theories are constructed based on evidence. In the sciences one rarely can prove the theory, but one accepts it because it explains the physical phenomena. However, it is always subject to change. If new evidence surfaces, the whole theory might change. In mathematics, it is very different since theories are proven, and with a correct proof they are true forever!
Student Learning Opportunities
1 (a) Give a direct proof that the exterior angle of a triangle is the sum of the two remote interior angles. That is, w = x + y. [Hint: The sum of the angles of a triangle is 180◦ as is the sum of z and w. See Figure 1.9 below.]
y
x z w
Figure 1.9
(b) As a corollary of (a), deduce that the exterior angle of a triangle is greater than either of the remote interior angles.
(c) As another corollary, prove by contradiction that, from a point outside a line, there can only be one perpendicular drawn to that line.
2 Give a direct proof that the figure with coordinates (5, 0), (3, 3), (−5, 0), and (−3, − 3) is a parallelogram.
3 Give a direct proof that 1 + 3 + 5. . . + (2n − 1) = n2 by showing that the sum is (1 + 2 + . . . + 2n) − (2 + 4 + 6 + . . . . + 2n) = (1 + 2 + . . . + 2n) − 2(1 + 2 + 3 + . . . . + n) and then using Theorem 1.1.
4 Using the facts from trigonometry that sin2θ + cos2θ = 1, and that cos 2θ = cos2θ − sin2θ, give a direct proof that cos 2θ = 1 − 2 sin2θ, and hence that sin2θ = 1 + cos 2θ for any angleθ. 2
5 (C) Students are asked to expand the expression (a + b)3. They do the computations and get a3+ 3a2b + 3ab2+ b3. How do they arrive at this expression? Is this a proof that (a + b)3= a3+ 3a2b + 3ab2+ b3? If so, what kind of a proof is it? Why?
6 Give a direct proof that 1
and so on. This pattern continues where the sum is always the middle number squared.
Using Theorem 1.1 see if you can explain the pattern. [Hint: The typical expression on the left can be written as 1 + 2 + 3 + . . . + n + n − 1 + n − 2 + . . . + 1.]
9 (C) Students often have trouble with proofs by contradiction. They don’t understand why when you negate an “if–then” statement, you assume the “if” part and negate the “then”
part. Show, using logic tables, that the negation of ( p→ q) is equivalent to (p∧ ∼ q).
Then explain how this equivalence is used as the basis for a proof by contradiction.
10 Give a proof by contradiction that, if 3n + 5 is even, then n must be odd.
11 Give a proof by contradiction that, if x + y< 12, then either x < 6 or y < 6.
12 Give a proof by contradiction to show that, if two lines l and m are cut by a transversal, in such a way that the alternate interior angles, x and y are equal, then the lines are parallel.
[Hint: If the lines aren’t parallel, then they meet at some point P as shown in the second picture of Figure 1.10 below. Now use Student Learning Opportunity 1 part (b).]
x y
13 True or False: When you give a proof by contradiction, you must contradict something that is given. Explain.
14 (C) A student asks if you can use the same method to prove√
3 is irrational as you used to show√
2 is irrational. How do you guide the student to see the differences and similarities in the proofs?
15 Give a proof by contradiction that 4 +√
3 is irrational. (You may need to use the fact that the difference of rational numbers is rational.)
16 Give a proof by contradiction that there cannot be a quadrilateral whose consecutive sides are A B = 2, B C = 3, C D = 5, and D A = 12. [Hint: Draw diagonal AC cutting the quadrilateral into two triangles. Using the fact that the shortest distance between two points is a straight line, show that the length of AC is less than 5. Now work with the other triangle. The shortest distance from A to D should be 12.]
17 Prove or disprove the following statement: “3n> n + 2 for each positive integer n.” Explain what method you used.
18 What method of proof was used to disprove Euler’s conjecture that there are no positive integers a, b, c, and d which make a4+ b4+ c4= d4?
19 Find a counterexample to the statement “The smallest natural number, n, such that the sum of the first n natural numbers is greater than 1000 is n = 50.”
20 (C) Your students are convinced that the following statement is true: x2− 1
x− 1 = x + 1. Are they correct? Give a proof for why this is or is not correct. What type of proof did you give?
21 (C) Your students are convinced that the following statement is true: If a< b then a2< b2. Are they correct? Give a proof for why this is or is not correct. What type of proof did you give?
22 (C) Your students have proven that, when they add 2 consecutive integers, they always get an odd number. Now they have begun to investigate what happens when then add 3 consecutive integers. Some have decided that the sum is always divisible by 3. Others have decided that the sum is always divisible by 6. Prove or disprove each of your students’
conjectures. Which method or methods did you use?
23 Prove or disprove: If three consecutive integers are multiplied together, and the second, in order of size, is added to the product, the result is always a perfect cube.
24 (C) A student asks, “ Since we can always use direct proof, why do we need to know proof by contradiction and proof by counterexample.” What is your reply?
25 (C) A student asks, “What happens when you try to prove that√
4 is irrational in a manner similar to the way we proved that √
2 was irrational? Won’t that same proof show that
√4 is irrational?” How do you explain to your student that the same method won’t work?
Where does the proof break down?