Just playing with numbers, arithmetic really. There are many ways of classifying numbers:
square nos, triangular nos, real nos, imaginary nos, rational nos, even irrational nos.
Most important for us today are whole numbers, aka “integers”.
In particular, the subset of whole nos which are ‘square nos’.
2. List square nos
It will be useful to have a list of some squ nos:
the nos 1–20 and build up a list of their squares – on
3. Tip for Next square no.
3. How to work out next square no: eg from 122 to 132 .
Add a row of 12 and a col of 12 and then 1 in the corner DO on paper, then show slide3
3B. BTW this neatly illustrates the expansion of (n + 1) 2 = n2 +2n + 1 Slide 4
3C. Finish the list of square nos. Possibly show 192 = 202 – 2n + 1.
4. Pythagoras’s Theorem –
In a RATriangle, the square on hypotenuse is equal to the sum of the squares on the other two sides. slide 5 - shows a,b,c
Example always used is 3, 4 5 – show it works slide6
‘What use is it?’ ; “used by builders” using knotted rope 12 feet long ...
Also I used it in my garden, to mark position of new driveway at rt angles to house wall.
5. Folded square
Must show you st surprising I discovered other day:
Square, A, B, C, D. Mark midpoint M of CD. Fold A to M. Label P (where fold occurs on AD).
Triangle MDP is rt angled triangle, and its sides are in ratio 3:4:5. Surprise!!
Measure to ‘prove’ it.
a ‘fact of the universe’,
a naturally occurring 3,4,5 tri! (like ratio of circ to diam is
always pi, a constant of the univ, it just IS)
Leave you to prove it at home, or try it with paper and ruler. NB is Ratio 3:4:5 not nec those nos.
6. [True for non-integers] [I will stick to integers]
Pythag is true for any plane RATriangle (eg 1.8inches, 2.4, 3.0).
But one branch of Mathematics deals only in domain of whole nos, Integers.
I want to stick to that, to whole numbers.
For ex, RATs where all 3 sides are whole nos, (like 3, 4, 5 units long)
Same as saying ... whole number solns to Pythag Equn: a2 + b2 = c2 slide7
Eg 3,4,5... Is called a Pythagorean Triple.
7. Most sets of 3 whole nos are Not PTs.
Let’s try some more: 4, 5, 6 [using flipchart of squ nos] [5, 6, 7] [4, 6, 7] So PT’s are special.
8. One, many, infinite?
poss that 3, 4, 5 is only; or maybe
a limited no; or an infinite no. Slide8
Mathematicians would ask (and try to prove or disprove):
Is there more than one solution?
If so, are theire many solutions?
If so, are there infinitely many solutions?
9. Suggest Triples?
anyone who’s new to this, or a bit rusty, like to suggest ano set
of 3 which might ‘work’.
Do 'discover' 6,8,10. If nec, draw as triangle by doubling a 3,4,5 tri and ask...
Where does this lead us...? many PT’s. Very many? Infinitely many! Write: [6,8,10]; [9,12,15];
Do discover 5,12,13.
10. [Primitive PT’s]
So: some/many/infinite no of PTs are just multiples of smaller PTs, they have an originator or grandaddy.
ones are of special interest,
Primitive PTs are ones where the three nos have no common factor: (Eg not all even...)
11. My Quest. Years ago I tried to find a way to invent more PTs (other than simple multiples of ones I already knew). (Before internet aval!) I came up with a method. I recently worked it out again:
Rewrite Pythag equn a2 + b2 = c2
as a2 = c2 - b2
Difference of two squares (from sch arith) ... =(c + b)(c - b)
at 5,12, 13 diff of one between 2 longer sides. So
Let (c – b) = 1
Then a2 = (c + b)
c – b = 1
Draw a table:
So we could take a squ no as a2, eg 49...
[work it thru, and prove]
Try 36... doesn’t ‘work’.
ph Try 64... doesn’t ‘work’.
[Formula, and Conditions??] [c, b] = ½ (a2 1) for odd a.
This is only for c - b = 1.
idea works for ...2, etc.. Leave to you to explore...
NB1 this also shows an infinite no... NB2 with diff of one, these are all PPT’s.
[[There are other ways (easier ways?) invented by other ppl.... the m, n method of Euclid...]]
X1. Pentagon on the hypotenuse...
Here’s something interesting that never occurred to me before I read it just the other day:
Pythag says: The square on hypotenuse equals sum of squares on other two sides.
Now imagine not a square, but some other shape (eg regular pentagon) on each side of the triangle. The pentagon on the hypotenuse equals sum of pentagons on other 2 sides!! (areas)
True for any shape, even irreg shape, eg shapes with curves (eg semicircle),
so long as they are methematically ‘similar shapes’.
...”The hump on the hypotenuse is equal to the sum of the humps on the other 2 sides!”
[eg In a 3, 4, 5 triangle the lengths of the sides are in ratio 3:4:5; so areas are in ratio 9:16:25.......]
X2. The Pentagon (enigma) puzzle... (What sparked my recent interest in PT’s was this little puzzle from New Scientist:
I have a rectangular piece of paper. From a point on one side I draw straight lines to the two adjacent sides (thus creating two triangles and a pentagon). All the sides are integers, and the sides of the pentagon are consecutive integers, not necessarily in order, and all less than 50. What size is the paper?
Hint: “it renewed my interest in Pythagorean Triples.”