Write the given number in the form a+i b. ((4+5 i)+2 i^3)/((2+i)^2)

Name: Problem 20, Chapter 17: Advanced Engineering Mathematics
Uploaded: 2022-01-10T14:10:51-08:00
Duration: 2 min 3 s
Channel: Yuva S
Description: Write the given number in the form a+i b. ((4+5 i)+2 i^3)/((2+i)^2)

step 1 In this question we will learn about some basic concept of the complex numbers. So in the given

Write the given number in the form a+i b. ((4+5 i)+2...

Key Concepts

Complex Numbers

Complex numbers are numbers that comprise a real part and an imaginary part. They are typically expressed in the form a + ib, where a is the real part and b is the coefficient of the imaginary part. This concept extends the idea of one-dimensional number lines to include a second, orthogonal dimension for imaginary numbers.

Imaginary Unit

The imaginary unit, denoted as i, is defined by the property that i² = -1. This definition is used to handle the square roots of negative numbers and underpins the arithmetic of complex numbers. Its cyclical powers (i, -1, -i, 1, …) are essential for simplifying expressions involving complex exponents.

Arithmetic Operations on Complex Numbers

Arithmetic with complex numbers involves addition, subtraction, multiplication, and division, following rules similar to real number arithmetic but with the introduction of i. When multiplying or dividing complex numbers, it is important to combine like terms and account for the fact that i² = -1, thereby converting any powers of i into real numbers when possible.

Rationalization Using the Complex Conjugate

When dividing complex numbers, rationalization is often done using the complex conjugate. The conjugate of a complex number a + ib is a - ib. Multiplying the numerator and the denominator by the conjugate of the denominator eliminates the imaginary part from the denominator, thus simplifying the expression into standard form.

Standard Form (Rectangular Form)

Expressing a complex number in standard or rectangular form means writing it as a + ib. This form clearly delineates the real and imaginary components, making it easier to interpret and further operate on the number. Converting expressions into this form is a fundamental skill in complex number arithmetic.

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