Problem 90 - A Graphical Approach to College Algebra

Question

Find the conjugate of each number. $$-3+i$$

Step-by-Step Solution

Answer

The conjugate is \(-3-i\).

Step 1: Understanding Conjugates

The conjugate of a complex number is obtained by changing the sign of the imaginary part. If we have a complex number of the form \(a + bi\), its conjugate will be \(a - bi\). In this case, our number is \(-3 + i\).

Step 2: Identifying Components

Write down the real part and the imaginary part of the given number. For \(-3+i\), the real part is \(-3\) and the imaginary part is \(i\).

Step 3: Applying Conjugate Definition

Change the sign of the imaginary part. So, the imaginary part \(+i\) becomes \(-i\).

Step 4: Writing the Conjugate

Combine the real part and the new imaginary part to form the conjugate. The conjugate of \(-3+i\) is \(-3-i\).

Key Concepts

Conjugate

The concept of a conjugate in complex numbers revolves around altering the sign of the imaginary part of a complex number. When you're given a complex number, say in the form

• \( a + bi \)

'identifying and changing this imaginary part is how you find its conjugate.
For instance, if you start with

• \(-3 + i\)

the imaginary portion, \(+i\), is adjusted to \(-i\).
This doesn't affect the real part of the number. Thus, the conjugate becomes

• \(-3 - i \).

The process involves a simple sign change but plays a critical role in various calculations, such as simplifying expressions or finding reciprocals of complex numbers.

Imaginary Part

The imaginary part of a complex number is what sets it apart from real numbers. In a complex number of the form

• \( a + bi \)

'b' is the coefficient of the imaginary unit 'i'.
For example, in the complex number

• \(-3 + i\)

the imaginary part is \(+i\) or \(+1i\).
The imaginary component is crucial when considering operations like finding conjugates or performing arithmetic with complex numbers.
This is because the imaginary part is the piece of the number that gets altered to find the conjugate, making its understanding pivotal.

Real Part

The real part of a complex number is essentially the portion that does not involve the imaginary unit.

• This part aligns with the numbers you might already know, broadly classed as 'real numbers.'

In a complex number expressed as

• \( a + bi \)

\('a'\) represents the real part.
Consider the example

• \(-3 + i\)

, where the real part is clearly

• \(-3\).

Unlike the imaginary part, the real part remains unchanged when finding the conjugate.
It serves as the constant anchor in complex numbers that provides a reference point around which calculations are performed.