Question

What is the reciprocal of `6+i`?

the answer is `(6-i)/37` but i dont know how. please explain!

Answer 1

`(6-i)/(37)`

Explanation:

`6+i`

reciprocal:

`1/(6+i)`

Then you have to multiply by the complex conjugate to get the imaginary numbers out of the denominator:

complex conjugate is `6+i` with the sign changed over itself:

`(6-i)/(6-i)`

`1/(6+i)*(6-i)/(6-i)`

`(6-i)/(36+6i-6i-i^2)`

`(6-i)/(36-(sqrt(-1))^2)`

`(6-i)/(36-(-1))`

`(6-i)/(37)`

Answer 2

The reciprocal of `a` is `1/a`, therefore, the reciprocal of `6+i` is:

`1/(6+i)`

However, it is bad practice to leave a complex number in the denominator.

To make the complex number become a real number we multiply by 1 in the form of `(6-i)/(6-i)`.

`1/(6+i)(6-i)/(6-i)`

Please observe that we have done nothing to change the value because we are multiplying by a form that is equal to 1.

You may be asking yourself; "Why did I choose `6-i`?".

The answer is because I know that, when I multiply `(a+bi)(a-bi)`, I obtain a real number that is equal to `a^2+b^2`.

In this case `a = 6` and `b=1`, therefore, `6^2+1^2 = 37`:

`(6-i)/37`

Also, `a+bi` and `a-bi` have special names that are called complex conjugates.