An algebra problem by Souvik Roy

Algebra Level pending

For integers a , b , c , d a,b,c,d , there exists 2 complex numbers a + b i a+bi and c + d i c+di such that their product is 1.

Check with proof whether any one of them is completely real or not.

If so, find its least value.

Not possible -1 1 3

Souvik Roy by Souvik Roy , 18, India

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1 solution

Parth Sankhe
Oct 19, 2018

We can take b and d to be 0, as we are only concerned with the real numbers.

a c = 1 ac=1 , with both a and c as integers. None can be greater 1 or lesser than -1, as this will force the other to be a fractional number, as their product is 1.

Thus the least value becomes − 1 -1 .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

We are asked to prove it. One method might be multiplying the two complex no.s and putting the product equal to 1. Now we can compare the real and imaginary parts on both sides of the equation. We would have ac-bd=1 and ad+bc=0. Square both of them, add and factorise. We get (...)*(...)=1. The terms in the brackets are all natural no.s. thus each term in the product is 1. Hence, all the no.s are>=0 and <=1. From this just put the values and find the least.

BTW, the proof isn't of much importance as it is an MCQ.

Souvik Roy - 2 years, 7 months ago

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