Maths Item of the Month
November 2010
1. On the same page, use Autograph to draw the graphs of
x 2 + y 2 + 8 xy − 1 = 0 and x 2 + y 2 − 8 xy − 1 = 0 .
Can you explain what is going on?
2. Now use Autograph to draw the graphs of
x + y + sqrt ( 8 mod ( xy ) ) − 1 = 0 and x 2 + y 2 − sqrt ( 8 mod ( xy ) ) − 1 = 0
Can you explain what is going on now?
2
2
Solution
1. Four circles
( x ± 1) + ( y ± 1)
2
( x − 1) + ( y − 1)
2
2
2
= 1 can be combined in one equation:
=1
Expanding this gives
x2 + y2 + 1 = 2 ( x + y )
Squaring:
(x
2
+ y 2 + 1) = 4 x 2 + 4 y 2 + 8 xy
Expanding and then factorising:
(x
2
+ y 2 − 1) = 8 xy
And this is the same as
2
2
x 2 + y 2 − 1 = ± 8 xy
2. If you draw the graph of y = 4 mod ( x ) you will realise that this is not the same as
y = 4 x as you might have expected.
Autograph is actually working with modulo arithmetic: y = 4 ( mod x ) . So, for
example, when x = 3 , y takes the value 4 ( mod 3) = 1 .
It follows that 8 mod ( xy ) = 1 will give all the curves xy =
7
for k = 1, 2,3, 4, 5, 6 .
k
But it also gives the curve xy = 8 . Why?
x 2 + y 2 ± sqrt ( 8 mod ( xy ) ) − 1 = 0 has more layers of complexity but you get the idea!
Thanks to Nick Lord for this Item.