Math T 1999 paper 1

[michiyo84]

Determine the value of a if (√2+ai)/(1+√2i) is a real number and find this real number.

How do I get the value a?

[Chelsea]

mind to tell what is the ? in between the two brackets. I think I can do this one if you tell me whats the ?

[michiyo84]

Oh sorry. It's the symbol of division. :)

[plawks]

http://latex.codecogs.com/gif.latex?\frac{\sqrt{2}+ai}{1+\sqrt{2}i}=%20c+di
where
http://latex.codecogs.com/gif.latex?c\in\mathbb{R},%20d=0
Multiplying both sides with http://latex.codecogs.com/gif.latex?1+\sqrt{2}i
http://latex.codecogs.com/gif.latex?\sqrt{2}+ai=c+c\sqrt{2}i
By comparing,
http://latex.codecogs.com/gif.latex?c=\sqrt{2}
http://latex.codecogs.com/gif.latex?a=c\sqrt{2}=\sqrt{2}\sqrt{2}=2

[michiyo84]

is that really the formula? i mean, C is the element of R and what R is actually? d is zero? why so?

[plawks]

There is no formula. Complex numbers can be expressed in the form of 'a+bi' where both a and b are real numbers. If the number is a real number then b=0.

I used 'c+di' so that the unknown 'a' used in the question is not repeated. Simple as that. Oh yea, 'ℝ' represents real numbers.

[michiyo84]

why would the b become zero when a is a real number? thank you for explaining by the way!! I appreciate it!

[Nicholasng925]

why would the b become zero when a is a real number? thank you for explaining by the way!! I appreciate it!

b will not become zero when a is a real number. Both could be non-zero, but in this case, the question was 'Determine the value of a if (√2+ai)/(1+√2i) is a real number and find this real number'.

We are talking about complex number, and we know that complex number comprises of real and imaginary part. Real part means value without 'i', while complex part means value with 'i', and both would form 'a+bi'.

Now back to the question, find the value of a if the whole expression is a REAL number, so already implied that the complex part is zero, in other word, b is 0. :)