What is a point of using complex numbers here?

  • #1
Hill
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Homework Statement
"Here is a basic fact about integers that has many uses in number theory: If two integers can be expressed as the sum of two squares, then so can their product. With the understanding that each symbol denotes an integer, this says that if ##M = a^2 + b^2## and ##N = c^2 + d^2##, then ##MN = p^2 + q^2##. Prove this result by considering ##|(a + ib)(c + id)|^2##."
Relevant Equations
##|x+iy|^2 = x^2 + y^2##
Firstly, the exercise itself is not difficult:
On one hand, $$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$
On the other hand, ##(a + ib)(c + id) = p+ iq## for some integers p and q, and so $$|(a + ib)(c + id)|^2 = |p + iq|^2 = p^2 + q^2.$$
Thus, ##MN = p^2 + q^2.##

However, it can be done quite straightforwardly, without considering the complex numbers, e.g.,
$$MN = (a^2 + b^2) (c^2 + d^2) = a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2 =$$$$a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2 + 2abcd - 2abcd = $$$$(a^2 c^2 + 2abcd + b^2 d^2) + (a^2 d^2 - 2abcd + b^2 c^2) =$$$$(ac + bd)^2 + (ad - bc)^2 = p^2 + q^2.$$

I don't see an advantage of considering the complex numbers in this case. What am I missing?
 
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  • #2
You could prove this slightly more easily by using the properties of the complex modulus. The required algebra is already encapsulated in the equation ##|zw|^2 = |z|^2|w|^2##.
 
  • #3
PeroK said:
You could prove this slightly more easily by using the properties of the complex modulus. The required algebra is already encapsulated in the equation ##|zw|^2 = |z|^2|w|^2##.
Thank you. I thought I've used it in this line:
$$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$
Is there something else there that I didn't use and that could simplify it further?
 
  • #4
Hill said:
Thank you. I thought I've used it in this line:
$$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$
Is there something else there that I didn't use and that could simplify it further?
That looks simpler than the alternative to me.
 
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FAQ: What is a point of using complex numbers here?

What is a point of using complex numbers in engineering?

Complex numbers are used in engineering, especially in electrical engineering, to simplify the analysis of alternating current (AC) circuits. They allow engineers to represent sinusoidal waveforms as phasors, making it easier to perform calculations involving impedance, current, and voltage.

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