Introduction

Lecture 1Introduction

Marlena Nowaczyk

Introduction

What is Mathematics?

Mathematics is the language of science and technology

• language is used for communication

• own vocabulary: points, lines, circles, velocity, functions,relations, transformations, equality, inequality...

• own alphabet: =,≤,√

2,P = (1, 0), x2 + y2 = 4,∂,∇, f (x) =,∀,∃, f ′, dSdt ,

∫f dx , . . .

• basic words: axioms

• grammar

• statements derived from axioms

• novel authors

Introduction

What is Mathematics?

Mathematics is the language of science and technology

• language is used for communication

• own vocabulary: points, lines, circles, velocity, functions,relations, transformations, equality, inequality...

• own alphabet: =,≤,√

2,P = (1, 0), x2 + y2 = 4,∂,∇, f (x) =,∀,∃, f ′, dSdt ,

∫f dx , . . .

• basic words: axioms

• grammar

• statements derived from axioms

• novel authors

Introduction

What is Mathematics?

Mathematics is the language of science and technology

• language is used for communication

• own vocabulary: points, lines, circles, velocity, functions,relations, transformations, equality, inequality...

• own alphabet: =,≤,√

2,P = (1, 0), x2 + y2 = 4,∂,∇, f (x) =,∀,∃, f ′, dSdt ,

∫f dx , . . .

• basic words: axioms

• grammar

• statements derived from axioms

• novel authors

Introduction

What is Mathematics?

Mathematics is the language of science and technology

• language is used for communication

• own vocabulary: points, lines, circles, velocity, functions,relations, transformations, equality, inequality...

• own alphabet: =,≤,√

2,P = (1, 0), x2 + y2 = 4,∂,∇, f (x) =,∀,∃, f ′, dSdt ,

∫f dx , . . .

• basic words: axioms

• grammar

• statements derived from axioms

• novel authors

Introduction

What is Mathematics?

Mathematics is the language of science and technology

• language is used for communication

• own vocabulary: points, lines, circles, velocity, functions,relations, transformations, equality, inequality...

• own alphabet: =,≤,√

2,P = (1, 0), x2 + y2 = 4,∂,∇, f (x) =,∀, ∃, f ′, dSdt ,

∫f dx , . . .

• basic words: axioms

• grammar

• statements derived from axioms

• novel authors

Introduction

What is Mathematics?

Mathematics is the language of science and technology

• language is used for communication

• own vocabulary: points, lines, circles, velocity, functions,relations, transformations, equality, inequality...

• own alphabet: =,≤,√

2,P = (1, 0), x2 + y2 = 4,∂,∇, f (x) =,∀, ∃, f ′, dSdt ,

∫f dx , . . .

• basic words: axioms

• grammar

• statements derived from axioms

• novel authors

Introduction

What is science?

Natural science:

• formulating equations (modelling)

• solving equations (computation)

What is the difference between “thinking” and “computing”?

Task 1. Find out which Nobel Prize Winners got the prize forformulating or solving equations.

Introduction

What is science?

Natural science:

• formulating equations (modelling)

• solving equations (computation)

What is the difference between “thinking” and “computing”?

Task 1. Find out which Nobel Prize Winners got the prize forformulating or solving equations.

Introduction

What is this course for?

• Do you like mathematics or hate mathematics, or somethingin between? Explain your standpoint.

• Specify what you would like to get out of your studies ofmathematics.

• In which form would you like to get out this?

Introduction

What is this course for?

• Do you like mathematics or hate mathematics, or somethingin between? Explain your standpoint.

• Specify what you would like to get out of your studies ofmathematics.

• In which form would you like to get out this?

Introduction

What is this course for?

• Do you like mathematics or hate mathematics, or somethingin between? Explain your standpoint.

• Specify what you would like to get out of your studies ofmathematics.

• In which form would you like to get out this?

Introduction

Definition, Theorem, Proof

A definition is a passage that explains the meaning of a term.A theorem is a statement derived from the axioms or othertheorems by using logical reasoning following certain rules of logic.Often uses construction “If . . . then . . . ” or “Let . . . then . . . ”The derivation is called a proof of the theorem.

The liar paradox:This sentence is falseThe barber paradox:A male barber shaves all and only those men who do not shavethemselves. Does he shave himself?

Introduction

Definition, Theorem, Proof

A definition is a passage that explains the meaning of a term.A theorem is a statement derived from the axioms or othertheorems by using logical reasoning following certain rules of logic.Often uses construction “If . . . then . . . ” or “Let . . . then . . . ”The derivation is called a proof of the theorem.

The liar paradox:This sentence is falseThe barber paradox:A male barber shaves all and only those men who do not shavethemselves. Does he shave himself?

Introduction

Pythagorean theorem

• State the Pythagorean theorem

• Prove the Pythagorean theorem

Task 2. Find other proofs of the Pythagorean theorem.

Introduction

Pythagorean theorem

• State the Pythagorean theorem

• Prove the Pythagorean theorem

Task 2. Find other proofs of the Pythagorean theorem.

Introduction

Pythagorean theorem

• State the Pythagorean theorem

• Prove the Pythagorean theorem

Task 2. Find other proofs of the Pythagorean theorem.

Introduction

Pythagorean theorem

• State the Pythagorean theorem

• Prove the Pythagorean theorem

Task 2. Find other proofs of the Pythagorean theorem.

Introduction

Some formulae

∫ ∞

0

1

1 + x2dx =

π

2

e iπ + 1 = 0

Introduction

Some formulae

∫ ∞

0

1

1 + x2dx =

π

2

e iπ + 1 = 0

Introduction

Never ever divide by zero!

1

x − 5, x ∈ [4, 10]

∞∑k=0

1

k

Introduction

Never ever divide by zero!

1

x − 5, x ∈ [4, 10]

∞∑k=0

1

k

Introduction

Never ever divide by zero!

1

x − 5, x ∈ [4, 10]

∞∑k=0

1

k

Introduction

Greek alphabet

A α

B

β

Γ

γ∆ δ

E ε ε Z ζ H η Θ θ ϑ

I

ι

K κ Λ λ M µ

N ν Ξ ξ O o Π

π

$

P ρ % Σ σ ς T τ Υ υ

Φ

φ

ϕ X χ Ψ ψ

Ω

ω

Introduction

Greek alphabet

A αB β Γ γ

∆ δ

E ε ε Z ζ H η Θ θ ϑ

I ι K κ Λ λ M µ

N ν Ξ ξ O o Π π $

P ρ % Σ σ ς T τ Υ υ

Φ φ ϕ X χ Ψ ψΩ ω

Introduction

Mathematical inductionTask 3. Find out who was the first one to give an explicitformulation of the principle of induction?Illustration of this principle is by the sequential effect of fallingdominoes.

Introduction

Proof by contradiction

Show that√

2 is an irrational number.