P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University...

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Proof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul 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.

Transcript of P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University...

Page 1: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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.

Page 2: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 3: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 4: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Visual Proof

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

1 + 3 + 5 + 7 + … (2n-1) = ?

Page 5: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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)

Page 6: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 7: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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?

Page 8: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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, …

Page 9: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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?

Page 10: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Visual ProofWhat is sum of the multiples of 5?

1st 2nd 3rd 4th … nth

5 + 10 + 15 + 20 + … 5n = ?

Page 11: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Visual ProofName other applications of this knowledge.

SeeA Triangular Journey

Page 12: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Powerful Visual ProofWhat is sum of the powers of 2?

1st 2nd 3rd 4th 5th … nth

1 + 2 + 4 + 8 + 16 … 2n-1 = ?

Page 13: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Solid ProofSome of the powers of 2 form a cube.

1 8 64 512 4096 327658 262144

Which ones are they?

Page 14: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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, …

Page 15: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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.

Page 16: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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.

Page 17: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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.

Page 18: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Try an Inductive Proof

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

Sum of even numbers is n(n+1)

Page 19: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

# 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.

Page 20: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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...

Page 21: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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...

Page 22: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

The nine digits problem

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

+=–=×=

Page 23: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

The nine digits problem

Here is one possible solution.

1 + 7 = 89 – 4 = 52 × 3 = 6

Is there another?

Page 24: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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?

Page 25: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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”.

Page 26: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 27: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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”)

Page 28: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 29: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 30: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Proof: the area of a circle is πr2

Page 31: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Proof: sum of interior angles of a triangle is 180

Page 32: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Proof: sum of pentagram angles is 180

Page 33: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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”!

Page 34: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 35: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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

Page 36: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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.

Page 37: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

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.

Page 38: P roof… Uniquely Mathematical and Creative Jim Hogan, School Support Services Waikato University Inspiration from Dr. Paul Brown, Carmel School, WA Paul.

Paul Brown can be contacted at [email protected]