Can you solve this complex number math Question?
Hello, I’m Teacher Jojo.
I teach mathematics in Korea.
Today, I have a question related to complex numbers.
It’s a question that is taught in the first year of high school in Korea.
If you know about complex numbers, give it a try and solve it.
[Can you solve this complex number math Question?]
What is the value of \; a² + b² \; for two real numbers a and b that satisfy the equation a+5i = \cfrac{2-bi}{1+i}\; ? ( i = \sqrt{-1})
…
…
…
Take some time to think about it and let’s see the solution below.
If you are familiar with complex numbers and the multiplication formula, you should be able to solve it.
…
…
…
[Solution]
Can you solve this complex number math Question?
To solve this question, we need to rationalize (I’m not sure if the term ‘rationalize’ is appropriate for changing the denominator into a real number instead of an imaginary number.) the denominator and utilize the concept of identities.
Let’s find the values of a and b.
Frist, Let’s try to rationalize the right-hand side.
\cfrac{2-bi}{1+i} \times \cfrac{1-i}{1-i} = \cfrac{(2-b)-(2+b)i}{2}
therefore,
a+5i = \cfrac{(2-b)-(2+b)i}{2}
thus,
a = \cfrac{2-b}{2} , \;\; 5 = \cfrac{-(2+b)}{2}
we can find the value of b.
b = -12
and we can find the value of a.
a = 7
a^2+b^2 = 7^2+(-12)^2 = 49 + 144 = 193
the answer is {\color{red}193}
How was today’s Question?
Did you find it interesting?
We needed to rationalize and find the values of a and b.
I hope you didn’t forget about the identity concept.
It would have been more fun if the problem involved using the multiplication formula, but it wasn’t the case, which is a bit disappointing.
I’ll come up with a more challenging problem next time.
I also have some previous posts you can check out.
Can you solve korea middle school math Question? #14
and if there’s anything you’re unsure about in your math studies, feel free to leave a comment.
I’ll be happy to help explain.
Well then, goodbye for now!
My youtube address https://www.youtube.com/channel/UCnJ-GLzfJdWjs04eQoxfY7g
[Korean ver]
안녕하세요. 저는 한국에서 수학을 가르치고있는 조조쌤입니다.
오늘은 복소수에 관한 문제를 가지고왔습니다.
이 문제는 고등학교 1학년 학생들이 푸는 문제입니다.
여러분도 도전해보세요!
[문제]
스스로 풀어보고 아래에 나와있는 풀이와 비교해보세요.
[해설]
우선 이 문제를 풀기위해서는 분모의 실수화를 해야합니다.
분모에 허수가 있으면 안되기 때문이죠.
그럼 분모의 실수화를 해볼까요?
분모, 분자에 1 – i를 곱해줍니다.
\cfrac{2-bi}{1+i} \times \cfrac{1-i}{1-i} = \cfrac{(2-b)-(2+b)i}{2}
그러므로
a+5i = \cfrac{(2-b)-(2+b)i}{2}
실수부분과 허수부분을 이용하여 식을 세울 수 있습니다.
a = \cfrac{2-b}{2} , \;\; 5 = \cfrac{-(2+b)}{2}
b값을 먼저 찾으면
b = -12
a값도 찾을수가 있습니다.
a = 7
a^2+b^2 = 7^2+(-12)^2 = 49 + 144 = 193
답은 {\color{red}193} 입니다.
오늘문제는 어땠나요?
비교적 쉬웠죠?
가끔 쉬운문제도 풀고 그래야죠.
다음에도 재미있는 문제로 찾아뵙겠습니다.
다음에 또봐요~
답글 남기기