Question

Simplify the multiplication of complex numbers: \[\left( {x,y} \right) \times \left( {1,0} \right)\]
A) \[\left( { - x, - y} \right)\]
B) \[\left( {y,x} \right)\]
C) \[\left( {x,y} \right)\]
D) None of these

Answer 1

Hint: We will solve this question by first taking \[\left( {x,y} \right)\] as \[\left( {x + iy} \right)\] and \[\left( {1,0} \right)\] as \[\left( {1 + i0} \right).\] Then on multiplying both the complex numbers we will get the required answer.

Complete step-by-step answer:
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i2 = −1. Because no real number satisfies this equation, i is called an imaginary number.

We have been given complex numbers and we need to simplify the multiplication \[\left( {x,y} \right) \times \left( {1,0} \right).\]
So, \[\left( {x,y} \right)\] can be written as \[\left( {x + iy} \right)\]
And \[\left( {1,0} \right)\] can be written as \[\left( {1 + i0} \right)\]
Now, we can write \[\left( {x,y} \right) \times \left( {1,0} \right)\] as \[\left( {x + iy} \right) \times \left( {1 + i0} \right)\]
So, \[\left( {x,y} \right) \times \left( {1,0} \right){\text{ }} = {\text{ }}\left( {x + iy} \right) \times \left( {1 + i0} \right)\]
\[\begin{array}{*{20}{l}}
  {\left( {x,y} \right) \times \left( {1,0} \right){\text{ }} = {\text{ }}\left( {x + iy} \right) \times \left( 1 \right)} \\
  {\left( {x,y} \right) \times \left( {1,0} \right){\text{ }} = {\text{ }}\left( {x + iy} \right)} \\
  {\left( {x,y} \right) \times \left( {1,0} \right){\text{ }} = {\text{ }}\left( {x,y} \right)}
\end{array}\]
So, the correct answer is “Option C”.

Note: Complex numbers are a combination of real and imaginary numbers. And when two complex numbers multiply, the first complex number gets multiplied by each part of the second complex number.