Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
MAT102 Intro to Math Proofs
Xinli Wang
University of Toronto Mississauga
January 9, 2019
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
Overview
1 Course InformationAssessment ComponentsHow to reach me?Course DescriptionImportant dates
2 Are they mathematical proofs?
3 Chapter 1: Numbers, quadratics and inequalitiesThe Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
4 Puzzle of the week
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
Assessment ComponentsHow to reach me?Course DescriptionImportant dates
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
Assessment ComponentsHow to reach me?Course DescriptionImportant dates
How can you find me?
In class.https://wangxinli.youcanbook.me/.Office Hours: Wednesday, 10AM-12PM, DH3060.Email me: [email protected] lunch with me!https://goo.gl/forms/FlMxI3xmdLAjrB2b2
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
Assessment ComponentsHow to reach me?Course DescriptionImportant dates
The goal of MAT102:
Understanding, using and developing precise expressions ofmathematical ideas, including definitions and theorems. Set theory,logical statements and proofs, induction, topics chosen fromcombinatorics, elementary number theory, Euclidean geometry.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
Assessment ComponentsHow to reach me?Course DescriptionImportant dates
Quiz and Term Test Dates
All quizzes and term test will happen on Thursdays, 6-7PM. Markthe following dates on your calendar:Jan 24th, Quiz 1Feb 7th, Quiz 2Feb 28th, Term TestMar 14th, Quiz 3Mar 28th, Quiz 4
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
https://www.maa.org/press/periodicals/convergence/
proofs-without-words-and-beyond-proofs-without-words-20
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
Prove: For any integer a, we have 2a2 > a.
Proof.
This proof is by case analysis. There are two cases:
Case 1: a is positive. Since a is an integer, we must have thata ≥ 1. Hence 2a2 = 2a · a ≥ 2a · 1 > a. This implies the claimholds in Case 1.
Case 2: a is negative. Since a is an integer, we must havethat a ≤ −1. Hence 2a2 ≥ 2 · (−1) · (−1) = 2 > −1 ≥ a.This implies the claim holds in Case 2.
The claim therefore holds in both cases.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Our very first theorem!
Thm 1.1.1 The Quadratic Formula
Let a, b, c be three real numbers, with a 6= 0. Then the equationax2 + bx + c = 0 has:
No real solutions if b2 − 4ac < 0.
A unique solution if b2 − 4ac = 0, given by x = − b2a .
Two distinct solutions if b2 − 4ac > 0, given by
x = −b+√b2−4ac2a and x = −b−
√b2−4ac2a .
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Basic Facts
Facts
1 if a < b or a ≤ b and c > 0, then ca < cb or ca ≤ cb.
2 a2 ≥ 0.
3 If a ≥ 0, then there is a unique nonnegative number√a
whose square is a.
4 If a < b and b < c, then a < c .
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Proposition 1.2.1
Let a and b be two real numbers.
1 If 0 < a < b, then a2 < b2 and√a <√b.
2 Similarly, if 0 < a ≤ b, then a2 ≤ b2 and√a ≤√b.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
What’s wrong?
Consider the following incorrect theorem:
Incorrect Theorem
Suppose that x and y are real numbers and x 6= 3. If x2y = 9ythen y = 0.
Proof.
Suppose that x2y = 9y . Then (x2 − 9)y = 0. Since x 6= 3, x2 6= 9,so x2 − 9 6= 0. Therefore we can divide both sides of the equation
(x2 − 9)y = 0
by x2 − 9, which leads to the conclusion that y = 0. Thus ifx2y = 9y , then y = 0.
What’s wrong with this proof? Can you find a counter example?Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Definition
The arithmetic mean of two real numbers, x and y , is x+y2 . If
x , y ≥ 0, then their geometric mean is√xy .
Check the following applet and make a conjecture about how thesetwo quantities are related:https://www.geogebra.org/m/rNv4xR5H
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Proposition 1.2.3(The Arithmetic-Geometric Mean Inequality).
For any two real numbers x and y ,
x · y ≤(x + y
2
)2
, (1)
and equality holds iff x = y . If, in addition, x , y ≥ 0, then
√xy ≤ x + y
2. (2)
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Application of Proposition 1.2.3 AGM
Use the AGM inequality to find the maximum of
(5 +√x4 + 1)(9−
√x4 + 1).
Prove that for any two real numbers x , y , with x 6= 0, we have
2y ≤ y2
x2+ x2.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Definition 1.3.1
The absolute value of a real number x , denoted as |x |, is defined as
|x | =
{x if x ≥ 0
−x if x < 0
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Proposition 1.3.2
For any two real numbers x , y , we have
√x2 = |x |, |x |2 = x2, x ≤ |x |, and |x · y | = |x | · |y |.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Proposition 1.3.3(The Triangle Inequality).
For any two real numbers x and y , we have
|x + y | ≤ |x |+ |y |.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Applications of the Triangle Inequality
Prove for any real number x and y ,
|x − y | ≥ |x | − |y |
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Applications of the Triangle Inequality
If |a| ≥ 2, |b| ≤ 12 . Find an M such that
|a + b| ≥ M.
What’s the maximum value of M?
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Example of triangle inequality
Let a, b, c be three real numbers. Prove that
|a− c | ≤ |a− b|+ |b − c |.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Definition 1.4.1
Let a be an integer, and b a nonzero integer. We say that a isdivisible by b (or that b divides a), if there exists an integer m, forwhich a = m · b.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Definition 1.4.2.
1 An integer is even if it is divisible by 2. Otherwise it is odd.
2 A natural number p > 1 is called a prime number,if the onlynatural numbers that divide p are 1 and p.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The Quadratic FormulaInequalities, and arithmetic/geometric meansThe Triangle InequalityType of Numbers
Exercise
Prove the following statement:There are infinitely many prime numbers.
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
An elderly shepherd died and left his entire estate to his three sons.To his first son, whom he favored the most, he bequeathed 1
2 hisflock of sheep, to the second son 1
3 , and to the third son, whom heliked the least, 1
9 of his flock.Not wishing to contest their father’s will, the three sons went tothe pasture to begin divvying up the flock. They were alarmed tocount a total of 17 sheep! Is there a means for the three sons tosuccessfully carry out their father’s wishes?
Xinli Wang MAT102/Week 1
Course InformationAre they mathematical proofs?
Chapter 1: Numbers, quadratics and inequalitiesPuzzle of the week
The End
Xinli Wang MAT102/Week 1