Question

If $z = 2 + i\sqrt { - 3} $, tell the real and imaginary part from standard form.

Answer 1

Hint:
We can simplify the negative inside the radical using i. Then we can write the standard form of a complex number. Then we can compare them and find the real and imaginary part. The term without i will be the real part and term with i will give the imaginary part.

Complete step by step solution:
We are given the complex number,
 $z = 2 + i\sqrt { - 3} $
We know that value inside the square root cannot be negative. So, we can expand the radicle as follows,
 $ \Rightarrow z = 2 + i \times \sqrt 3 \times \sqrt { - 1} $
We know that the imaginary term i is the square root of negative one. So, we can write it as $i = \sqrt { - 1} $ . On substituting this in z, we get,
 $ \Rightarrow z = 2 + i \times \sqrt 3 \times i$
Hence, we have,
 $ \Rightarrow z = 2 + {i^2} \times \sqrt 3 $
We know that $i = \sqrt { - 1} $ . On taking the square, we get, \[{i^2} = - 1\] . Then z will become,
 $ \Rightarrow z = 2 + \left( { - 1} \right)\sqrt 3 $
Hence, we have,
 $ \Rightarrow z = 2 - \sqrt 3 $
We know that the standard form of a complex number is given by $z = x + iy$ where x is the real part and y is the imaginary part.
So, we can write z as,
 $ \Rightarrow z = \left( {2 - \sqrt 3 } \right) + 0i = x + iy$
The real part is x. It is given by,
 $ \Rightarrow x = \left( {2 - \sqrt 3 } \right)$
The imaginary part is y. From the equation we get,
 $ \Rightarrow y = 0$

The given complex number in its standard form is given by $z = \left( {2 - \sqrt 3 } \right) + 0i$ where $\left( {2 - \sqrt 3 } \right)$ is the real part and 0 is the imaginary part.

Note:
We know that a complex number is defined as the ordered pair $\left( {x,y} \right)$ such that $z = x + iy$ where x and y are real numbers and i is the imaginary term which is given by $i = \sqrt { - 1} $ . We cannot take the given form of the complex number as a standard form as $\sqrt { - 3} $ is not a real number. After converting into standard form, we must note that the term with i is only considered as the imaginary part. As there is no term with i, the imaginary part will be zero.