+0  
 
0
94
1
avatar+169 

Ramanujan and Hardy played a game where they both picked a complex number. If the product of their numbers was 32 - 8i, and Hardy picked 5 + 3i, what number did Ramanujan pick?

matth99  Jul 2, 2018
 #1
avatar
+1

Simplify the following:
(-8 i + 32)/(3 i + 5)

Factor 8 out of 32 - 8 i giving 8 (4 - i):
(8 (-i + 4))/(3 i + 5)

Multiply numerator and denominator of (8 (-i + 4))/(3 i + 5) by 5 - 3 i:
(8 (-i + 4) (-3 i + 5))/((3 i + 5) (-3 i + 5))

(5 + 3 i) (5 - 3 i) = 5×5 + 5 (-3 i) + 3 i×5 + 3 i (-3 i) = 25 - 15 i + 15 i + 9 = 34:
(8 (-i + 4) (-3 i + 5))/34

The gcd of 8 and 34 is 2, so (8 (-i + 4) (-3 i + 5))/34 = ((2×4) (-i + 4) (-3 i + 5))/(2×17) = 2/2×(4 (-i + 4) (-3 i + 5))/17 = (4 (-i + 4) (-3 i + 5))/17:
(4 (-i + 4) (-3 i + 5))/17

(4 - i) (5 - 3 i) = 4×5 + 4 (-3 i) - i×5 - i (-3 i) = 20 - 12 i - 5 i - 3 = 17 - 17 i:
(4 -17 i + 17)/17

Factor 17 out of 17 - 17 i giving 17 (1 - i):
(4×17 (-i + 1))/17
Combine powers. (4×17 (-i + 1))/17 = 4×17^(1 - 1) (-i + 1):
4×17^(1 - 1) (-i + 1)

1 - 1 = 0:
4×17^0 (-i + 1)

17^0 = 1:
4 (-i + 1) = 4 - 4i - Ramanujan's number.

Guest Jul 2, 2018

19 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.