Lesson Objective Understand what maths is Understand some of the common notation attached to mathematics

Galileo GalileiGalileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth." Carl Friedrich GaussCarl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences". “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” S. Gudder “Maths is about spotting patterns.” M. de Sautoy Is Maths a language, a science, an art or a philosophical ideal? Is maths real or abstract? Do we invent mathematics or discover mathematics? Is maths creative, or merely right or wrong? Albert EinsteinAlbert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." [ is What is MATHS ?

A language: The purpose of a language is to describe and express the world we live in – maths does this so well and so universally that if we open a Maths text book in Russia you will likely understand the algebraic content as well as if you were in England. A Science: Science is about spotting patterns, making a theory and aiming to see if they are true by disproving them. When you are confident that they are true you build new theories. An Art: Maths is not about being right or wrong. It is a creative subject where you need vision and ‘blue sky’ thinking to connect ideas together. Many of the greatest discoveries are so elegant that they inspire awe, excitement and wonder. A philosophical ideal: Maths is entirely abstract, yet gives a vision of the real world which is strikingly familiar – in many subject areas where scientists were scared to follow, mathematics lead the way. Maths is the only subject where you true knowledge is obtained, it is fundamentally based on logic and truth.

The History of numbers

How maths Works: At the turn of the 19 th century people tried to ‘systemise’ mathematics by setting up basic axioms. These were universal basic truths from which the rest of mathematics could be derived. Most of the geometric axioms come straight from Euclid’s Elements Most of the other axioms come from basic ‘set theory’ In maths, we commonly spot something, make a conjecture and then aim to prove the conjecture using truths already proven. When a conjecture has been satisfactorily proved it becomes a Theorem. It can then be used to prove other conjectures. There are several ways of proving a conjecture, the most common of which are Direct argument/Logical deduction Exhaustion Contradiction Induction

 implies B We say that A is a sufficient condition for B. (It is sufficient for A to be true for B to be true) We can also say that B is a necessary condition of A. (If A is true it is necessary for B to be true)  implies A Notice that now B is a sufficient condition for A and A a necessary condition for B  iff (if and only if) or a double implication: A implies B and B implies A A and B are both sufficient and necessary conditions for each other Eg x = 3  x   We might say: It is sufficient for x = 3 in order for x 2 = 9 Or It is necessary for x 2 = 9 when x =3 But it is not necessary for x = 3 in order for x 2 = 9 Eg x + 7 = 15  x  We might say: It is sufficient for x + 7 = 15 in order for x  Or It is necessary for x  8 when  x + 7 = 15 It is also necessary for x + 7 = 15 when x = 8 Or It is sufficient for x =8 in order for x + 7 = 15 Eg A: animal is a spiderB: animal has 8 legs   (it is sufficient for A to be true for B to be true, but not necessary)

Other Notation you might see

Pick two numbers. Call one x and the other y Find: x + y y – x xy x/y y/x

x + y, y – x, xy, x/y and y/x

c ab ba a b b a c What does this show? How does it show it?

Lesson Objective: Understand the importance of proof in mathematics Look at proof by direct argument/logical deduction

Proof by direct argument is the easiest and simplest kind of proof to understand in mathematics. When proving things by direct argument we often use the symbols:    Start with something that is true Do indisputably true stuff to it End up with new stuff that must also be true

Eg Prove that the angle at the centre of a circle is half the angle at the circumference A X Y C

Consider the diagram opposite Triangle AXC is isosceles  angle XAC = angle AXC Triangle AYC is isosceles  angle YAC = angle AYC Angles in a triangle sum to 180 degrees  angle ACX = 180 – 2XAC Angles in a triangle sum to 180 degrees  angle ACY = 180 – 2YAC Angles at a point sum to 360 degrees  angle XCY = 360 – (180 –2YAC) – (180 –2XAC) = 2YAC + 2XAC q.e.d. Eg Prove that the angle at the centre of a circle is half the angle at the circumference A X Y C

Eg Prove: a)That the sum of two even numbers is even b) That the product of an even number and an odd number is even a)If a number is divisible by 2, then so is its square Prove: 1)The sum of two odd numbers is even 2)The difference between any two odd numbers is even 3)The product of two even numbers is even 4)The product of two odd numbers is odd 5)The product of an even number and an odd number is even 6)The sum of three consecutive integers is always a multiple of 3 7)The sum of two consecutive odd numbers is divisible by 4 8)If x is an odd number the difference between x 3 and x is always even

Direct argument proofs: 1)Prove that the sum of two consecutive squares is always odd 2)Prove that (a + b) 2 – (a – b) 2 = 4ab 3)If n is a positive integer, show that n 2 + n is always even. 4)Factorise n 3 + 3n 2 + 2n. Hence prove that, when n is a positive integer, n 3 + 3n 2 + 2n is always divisible by 6. 5)i) Prove that 12 is a factor of 3n 2 + 6n for all positive integers, n. ii) Determine whether 12 is a factor of 3n 2 + 6n for all positive integers, n. 5)Prove that if T is a triangular number then 8T +1 is always a square number. 6)Prove that the difference between odd squares is divisible by 8. 7)Prove that n 3 - n is always even, then extend your proof to show that it is always divisible by 6. Prove also that n 5 – n is divisible by 30.

More direct argument proofs: 8) Prove that the opposite angles in a cyclic quadrilateral sum to 180 degrees. 9) Prove that in a right angle triangle, if b and c differ by 1, then: b a c 10) Prove that (sin x + cos x) 2 – 1 = 2sin x cos x 11) Prove that

Lesson Objective: Understand how to prove things by exhaustion Understand how to prove things by contradiction In this diagram each region needs to be coloured so that no two adjacent regions have the same colour. What is the fewest number of colours I will need?

Proof by exhaustion Prove that 3 n is always an odd number

More exhaustion proofs 1)Prove that when a two digit number is divisible by 3, reversing its digits will also give a number divisible by 3. 2)No square number ends in an 8. 3)If ‘m’ is an even square number then is even for all ‘m’ less than 100. Can you extend your proof using a mixture of direct and exhaustion to prove that this result is true even if ‘m’ is not less than 100?

Assume something is true Then this will be true Do some maths that we know is definitely true Then this will be true Do some maths that we know is definitely true Then this will be true – but it can’t be! So something must be wrong. The original assumption must be wrong! Proof by contradiction

Eg Prove that is irrational.

Eg Prove that there is no smallest number bigger than 0

Silly Proof! Eg. Prove that all numbers are ‘interesting’.

Eg Prove that there are no positive integers, x and y, such that x 2 – y 2 = 1

Consider the following statement: When Fred Lemming is happy he eats chocolate: Fred Lemming is happy He is eating chocolate This is the same as saying If Fred isn’t eating chocolate he can’t be happy Fred isn’t eating chocolate he can’t be happy Proof by contra-positive

We can summarise this as follows: If we wish to show that A B It is often easier to prove the contrary statement that B being False A is False Note some people look at this as proof by contradiction still because: If you Assume that A is true, when B is false getting a contradiction would prove the result. Personally I think that this is rubbish and that proof by contra- positive is a separate case in its own merit!

Eg Prove that if n 3 is odd (where n is a +ve integer) then n is odd Instead prove that the converse statement is true: That when n is even n 3 is even

Eg Prove that if x is irrational then is irrational Instead prove that the converse statement that When is rational, x is rational.

Eg Prove that: 1)If a, b and c are integers such that bc is not divisible by a then b is not divisible by a 2)If (n 2 – 4n) is odd, when n is a positive integer, then n is odd. 3)There are no positive integers, m and n such that m 2 – n 2 = 6.

Disproving things This is generally a whole lot easier. We can either 1) Show by a logical argument Eg Disprove the statement: “4 is a solution to the equation x 2 +x(x – 3) = 7” Eg Disprove the statement: “(x – 3) is a factor of x 3 + 2x + 5 Eg Disprove the statement: “All quadratics have 2 real solutions”

OR 2) Give a case that doesn’t work – what we call a counter example Eg Disprove the statement: “Girls can’t do maths” Eg Disprove the statement: “All prime numbers are even” Eg Disprove the statement: “All quadratics have 2 real solutions”

Disprove the following: 1)For all positive real values of x. 2)For all positive integer values of n, (n 3 – n + 7) is prime. 3)15x 2 –11x + 2 ≠ 0 for all real values of x. 4)The difference between the squares of consecutive odd numbers is never divisible by 8. 5)There are no prime numbers divisible by 7. 6)If x and y are irrational and x > y then xy is irrational. 7) is irrational for all positive integers n. 8)If a, b and c are integers such that a is divisible by b and b is divisible by c then a is divisible by c.