We will learn how to expand a complex in the standard form a + ib.
The following steps will help us to express a complex number in the standard form:
Step I: Obtain the complex number in the form \(\frac{a + ib}{c + id}\) by using fundamental operations of addition, subtraction and multiplication.
Step II: Multiply the numerator and denominator by the conjugate of the denominator.
Solved examples on complex number in the standard form:
1. Express \(\frac{1}{2 - 3i}\) in the standard form a + ib.
Solution:
We have \(\frac{1}{2 - 3i}\)
Now multiply the numerator and denominator by the conjugate
of the denominator i.e., (2 + 3i), we get
= \(\frac{1}{2 - 3i}\) × \(\frac{2 + 3i}{2 + 3i}\)
= \(\frac{2 + 3i}{2^{2} - 3^{2}i^{2}}\)
= \(\frac{2 + 3i}{4 + 9}\)
= \(\frac{2 + 3i}{13}\)
= \(\frac{2 }{13}\) + \(\frac{3}{13}\)i, which is the required answer in a + ib form.
2. Express the complex number \(\frac{1 - i}{1 + i}\) in the standard form a + ib.
Solution:
We have \(\frac{1 - i}{1 + i}\)
Now multiply the numerator and denominator by the conjugate of the denominator i.e., (1 - i), we get
= \(\frac{1 - i}{1 + i}\) × \(\frac{1 - i}{1 - i}\)
= \(\frac{(1 - i)^{2}}{1^{2} - i^{2}}\)
= \(\frac{1 - 2i + i^{2}}{1 + 1}\)
= \(\frac{1 - 2i - 1}{2}\)
= \(\frac{- 2i }{2}\)
= - i
= 0 + (- i), which is the required answer in a + ib form.
3. Perform the indicated operation and find the result in the form a + ib.
\(\frac{3 - \sqrt{- 49}}{2 - \sqrt{-36}}\)
Solution:
\(\frac{3 - \sqrt{- 49}}{2 - \sqrt{-36}}\)
= \(\frac{3 - 7i}{2 - 6i}\)
Now multiply the numerator and denominator by the conjugate of the denominator i.e., (2 + 6i), we get
= \(\frac{3 - 7i}{2 - 6i}\) × \(\frac{2 + 6i}{2 + 6i}\)
= \(\frac{(3 - 7i)(2 + 6i)}{2^{2} - 6^{2}i^{2}}\)
= \(\frac{6 + 18i - 14i - 42i^{2}}{4 + 36}\)
= \(\frac{6 + 4i + 42}{40}\)
= \(\frac{48 + 4i}{40}\)
= \(\frac{48 }{40}\) + \(\frac{4}{40}\)i,
= \(\frac{6 }{5}\) + \(\frac{1}{10}\)i, which is the required answer in a + ib form.
11 and 12 Grade Math
From Complex Number in the Standard Form to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Aug 10, 24 04:59 PM
Aug 10, 24 02:25 AM
Aug 10, 24 01:59 AM
Aug 10, 24 01:57 AM
Aug 06, 24 02:12 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.