Numbers in my sleep
- Last night I had the strangest dream.
- Please share!
- I dreamt of two numbers whose product and sum were both one.
- How unusual!
- Yes, indeed.
- Did you dream what the numbers were?
- No. But I found the difference in my sleep.
- What was the difference?
June 11th, 2010 at 8:01 am
1. ab=1
b=1/a
2. a+b=1
a+1/a=1
(a^2)/a + 1/a = 1
a^2+1=a
a^2-i^2=a
(a+i)(a-i)=a
Case 1:
(a+i)/a=0
a=-i
ab=1
-ib=1
b=i
Case 2:
(a-i)/a=0
a=i
ab=1
ib=1
b=-i
So the terms are i and -i.
i – (-i) = 2i
The difference is 2i.
June 11th, 2010 at 8:02 am
While I can’t find the error in my math, there is certainly an error… i + -i = 0 (not 1, as required). Back to the drawing board.
June 11th, 2010 at 8:48 pm
How to find |x-y| when you know xy and x+y?
(x+y)^2 = x^2 + 2xy + y^2
and
(x-y)^2 = x^2 – 2xy + y^2
In other words:
(x-y)^2 = (x+y)^2 – 4xy = 1^2 – 4 = -3
so |x-y| = sqrt(3)i
June 12th, 2010 at 1:21 am
ab = 1
a + b = 1
a + 1/a = 1
a^2 + 1 = a
a^2 – 1 + 1 = 0
a = (1 +/- sqrt(3))/2
so:
a = 1/2 + i*sqrt(3)/2
b = 1/2 – i*sqrt(3)/2
a – b = i*sqrt(3)