Question

Divide and simplify. $$(5-6 i) \div(-3+2 i)$$

Short answer

ewline $$\frac{5-6i}{-3+2i} = -\frac{27}{13} + \frac{8}{13}i.$$

Step-by-step solution

Apply Complex Division

To divide by a complex number, multiply the numerator and denominator by the complex conjugate of the denominator.

Find the Complex Conjugate of the Denominator

The complex conjugate of the denominator ewline (-3+2i) is (-3-2i).

Multiply Numerator and Denominator by the Conjugate

Multiply (5-6i) and (-3-2i) as well as the denominator (-3+2i) with the conjugate (-3-2i).

Perform the Multiplication

Expand the products to get ewline Numerator: \((5-6i)(-3-2i) = -15 -10i + 18i + 12i^2\), Denominator: \((-3+2i)(-3-2i) = 9 - 4i^2\).

Simplify the Squares of i

Simplify by noting that \(i^2 = -1\) and thus getting ewline Numerator: \(-15-10i+18i-12\), Denominator: \(9-(-4)\).

Combine Like Terms

Combine like terms in both numerator and denominator ewline Numerator: \((-15-12) + (18i-10i) = -27 + 8i\), Denominator: \(9+4 = 13\).

Write the Result as a Complex Number

The quotient is the simplified complex number ewline $$\frac{-27 + 8i}{13} = -\frac{27}{13} + \frac{8}{13}i.$$

Key concepts

Complex Conjugate

When dividing complex numbers, one essential tool is the complex conjugate. For any complex number, such as \(a + bi\), its complex conjugate is \(a - bi\). The conjugate has the same real part, \(a\), but the sign of the imaginary part, \(bi\), is flipped.

The conjugate is particularly useful because when a complex number is multiplied by its conjugate, the result is always a real number. The imaginary parts cancel out, as \(i\) times \(i\) is \(i^2\), which equals \( -1\). Hence, \((a + bi)(a - bi) = a^2 - abi + abi - b^2i^2 = a^2 + b^2\), since \(b^2i^2\) equals \(b^2(-1)\).

Using the conjugate to divide by a complex number, as shown in the exercise, resolves the problem of having a complex denominator. It produces a real denominator, enabling the formulation of the result as a standard complex number: a real part plus an imaginary part.

Simplifying Complex Expressions

The simplification of complex expressions is a process that involves several algebraic manipulations. After performing operations such as multiplication or division, it's vital to consolidate the expression to its simplest form. This often means combining like terms and applying the definition that \(i^2 = -1\).

As exhibited in the provided solution, after expanding the product of complex numbers, one should look for real terms to combine and imaginary terms to consolidate independently. For example, \( -15 -10i + 18i + 12i^2 \) simplifies when \(12i^2\), which is \(12(-1)\), is recognized as \( -12\), a real number. This illustrates the importance of recognizing and applying the fundamental properties of \(i\) to simplify complex expressions effectively.

Complex Numbers Arithmetic

Arithmetic with complex numbers includes operations such as addition, subtraction, multiplication, and division, similar to real numbers but with the additional consideration of the imaginary unit \(i\).

When performing these operations, especially multiplication and division, it's crucial to apply the distributive property, also known as the FOIL method for binomials. In the context of our exercise, during multiplication, each term of the first complex number is multiplied by each term of the second, keeping in mind that \(i^2 = -1\).

In division, as the solution demonstrates, employing the complex conjugate of the denominator allows for the operation to be completed without leaving any imaginary numbers in the denominator. After the multiplication, the real and imaginary terms are separated, and any instances of \(i^2\) are simplified. The goal is always to express the final answer in the form \(a + bi\), where \(a\) and \(b\) are real numbers, thus facilitating easier interpretation and further calculations with the complex number.