Proof…Uniquely

Mathematical and CreativeJim Hogan, School Support Services

Waikato University

Inspiration from Dr. Paul Brown, Carmel School, WAPaul presented a similar workshop at MAV Conference,

2008. I enjoyed the session so much I decided then to present my version here in Palmerston North. Thanks also to a little book called Q.E.D. Beauty in Mathematical Proof

written by Burkard Polster.

See last slide for all contact details.

What will we learn today?

• Well, who knows? Proof is the pudding!

• There are a few problems to ponder• We might experience joy of proof• You might find a useful resource• Your students might benefit• Aspects of the NZC may be made

clear• and we must do something with

π

Proof

It is nice to experience surprise!

Like a good cryptic clue

14. Clothes appear wrong when put on a number (7)

P P

L L

Visual Proof

What is the sum of n odd numbers?1st 2nd 3rd 4th … nth

1 + 3 + 5 + 7 + … (2n-1) = ?

Visual Proof

What is the sum of n even numbers?

1st 2nd 3rd 4th … nth

2 + 4 + 6 + 8 + … 2n = ?

This presents a nice opportunity to show the square and a side; n2 + n = n(n + 1)

Algebra Re-Vision

A meaning of 1, n and n + 1.

What does 1 look like?What does n look like?What does n2 look like?What does n3 look like?

0 1 n n + 1

n - 1

Visual Proof

What is the sum of n counting numbers?

2 + 4 + 6 + 8 + … 2n = n2 + n1 + 2 + 3 + 4 +… n = ?

Notice that halving the even numbers makes the counting numbers.

How does that help?

Visual Proof

Make the triangular numbers

1st 12nd 1 + 23rd 1 + 2 + 34th 1 + 2 + 3 + 4 and so on…

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 425, …

Visual ProofTwo of the same triangular

numbers

make n2 + n = n(n + 1)So one of them is n(n + 1)/2

What is a triangular number?

Visual ProofWhat is sum of the multiples of 5?

1st 2nd 3rd 4th … nth

5 + 10 + 15 + 20 + … 5n = ?

Visual ProofName other applications of this knowledge.

SeeA Triangular Journey

Powerful Visual ProofWhat is sum of the powers of 2?

1st 2nd 3rd 4th 5th … nth

1 + 2 + 4 + 8 + 16 … 2n-1 = ?

Solid ProofSome of the powers of 2 form a cube.

1 8 64 512 4096 327658 262144

Which ones are they?

An odd look at numbers

nth odd is 2n -1

1 + 3 + 5 + 7 + … (2n-1)Notice (2n-1) = (n-1) + n. The odd numbers are consecutive

pairs

0+1, 1+2, 2+3, 3+4, 4+5, 5+6, …

Proof by Induction

The idea is easy.Prove the first is true.Show for any one k… the truthProve for the next one, k + 1…

the truthBy a dominoe effect if 1 is true

then so is 2 and 3 and 4 and 5 and …so on.

Proof by InductionOdd numbers are consecutive pairs0+1, 1+2, 2+3, 3+4, 4+5, 5+6, …

nth odd number = (n – 1) + n

Eg, n = 3, makes 3 – 1 + 3 = 2 + 3 = 5When n = k, this makes k – 1 + k = 2k - 1Put n = k + 1, makes k + 1 – 1 + k + 1 =

2k + 1Notice 2k+1 = 2k-1 +2 is next odd

number. This is INDUCTION.

Prove the sum of the odd numbers is n2.

When n = 1, n2 is 1When n = k, n2 is k2

When n = k + 1, k2 is (k+1)2

(k+1)2 = k2 + 2k + 1 =k2 + 2k – 1 +2

We need to see that 2k – 1 +2 is the next odd number added on… Q.E.D.

Try an Inductive Proof

Sum of whole numbers is n(n+1)/2

Sum of even numbers is n(n+1)

# of Vertices = # of Sides

What happens when you cut off one of the vertices of a triangle?

What the students may not realise is that they have performed a proof by induction.

pi time

In a circle of diameter 1, draw a square.

€

1

2€

1

2

€

1

2

The perimeter πd of the circle is approximated by the perimeter of the square

€

4 ×1

2= 2 2

€

π ×1= 2.8284...

pi time again

In a circle of diameter 1, draw an octagon.

€

1

2

€

1

2

The perimeter πd of the circle is approximated by the perimeter of the octagon

€

8 ×1

4+1

4−2 ×

1

2×1

2×cos45

= 4 × 2 − 2

€

π ×1= 3.0614...

The nine digits problem

The digits 1 to 9 have to go into the nine boxes, with no repeats.

+=–=×=

The nine digits problem

Here is one possible solution.

1 + 7 = 89 – 4 = 52 × 3 = 6

Is there another?

Here are a few solutions

And digits can be swapped between equations.1 + 7 = 8 7 + 1 = 8 4 + 5 = 9 4 + 5 = 99 – 4 = 5 9 – 4 = 5 8 – 7 = 1 8 – 1 = 72 × 3 = 6 2 × 3 = 6 2 × 3 = 6 2 × 3 = 6

Why is 2 x 4 = 8 not an option?

The nine digits problem

If three even digits are used in the multiplication, there are not enough even digits left for the addition and the subtraction.

E x E = EO + O = EO – O = uh uh

This is called “parity checking”.

That’s it Folks

This file is on my websitehttp://schools.reap.org.nz/advisor

Clearly labelled for you.There are a few extra files from Paul and references to his website and new book. Thank you…….jim

We could make a conjecture(Latin “throw together”)

Or could experiment with a supposition(Latin “place under” & “location”)

Or form an hypothesis(Greek “under” & “foundation”)

n2 is the sum of the first n odd numbers

n 1 2 3 4 … k k + 1

2n – 1 1 3 5 7 … 2k - 1 2(k + 1) - 1

n2 1 4 … k2 ??

?? = k2 + 2(k+ 1) - 1

= k2 + 2k+ 2 - 1

= k2 + 2k + 1

= ( k + 1)2

n2 is the sum of the first n odd numbers

n 1 2 3 4 … k

2n – 1 1 3 5 7 … 2k - 1

n2 1 4 …

The total of the middle row is 1 + 3 + 5 + … + 2k-5 + 2k-3 + 2k-1

= 1 + 2k-1 + 3 + 2k-3 + 5 + 2k-5 + …

= 2k + 2k + 2k + …

= 2k × k/2 = k2

Proof: the area of a circle is πr2

Proof: sum of interior angles of a triangle is 180

Proof: sum of pentagram angles is 180

Answers to other “Challenges”A. For any two numbers, the sum of their squares is never less than twice

their product.

(a – b)2 0

a2 – 2ab + b2 0

a2 + b2 2ab

Some people like to write Q.E.D. to mark the end of the proof. It is Latin, quod erat demonstrandum and means “Hooray! I’ve finished the proof!”.

What Q.E.D. does not stand for is “Quite Easily Done”!

Answers to other “Challenges”B. The sum of any positive number and its reciprocal is 2 or greater.

This is from the excellent “Proofs without words: Exercises in visual thinking” by R. B. Nelson (Mathematical Association of America)

Answers to other “Challenges”C. If the final score is a draw, the number of possible half-time scores is a square.

This is from the “Squares” CD-ROM.

Answers to other “Challenges”D. The product of any two Hilbert numbers is a Hilbert number.

The Hilbert numbers are 1, 5, 9, 13, 17, 21, 25, ... They are the numbers one greater than numbers in the four times table.

For n = 1, 2, 3, 4, 5, 6, ... the Hilbert numbers are 4n + 1.

Let the two numbers be 4x + 1 and 4y + 1 where x and y are natural numbers.

The product = (4x + 1) (4y + 1)

= 16xy + 4x + 4y + 1

= 4(4xy + x + y) + 1 which is a Hilbert number.

Paul Brown can be contacted at pbperth@gmail.com