Question

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. \(\frac{3 i-7}{i+4} \quad\) Carefull Not \(3-7 i\)

Short answer

The simplified form is \(\frac{31}{17} - i\frac{21}{17}\).

Step-by-step solution

Rationalize the denominator

To simplify \(\frac{3i-7}{i+4}\), start by multiplying the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(i+4\) is \(-i+4\).

Multiply the fractions

Multiply both the numerator and denominator by \(-i+4\): \[ \frac{(3i-7)(-i+4)}{(i+4)(-i+4)} \]

Distribute in the numerator

Use the distributive property to expand the numerator: \[ (3i-7)(-i+4) = 3i(-i) + 3i(4) - 7(-i) - 7(4) = -3 - 12 - 7i + 28i = -31 + 21i \]

Simplify the denominator

Expand and simplify the denominator using the difference of squares formula: \[ (i+4)(-i+4) = (i^2 - (4)^2) = -1 - 16 = -17 \]

Combine results

Combine the simplified numerator and denominator: \[ \frac{-31+21i}{-17} = \frac{31}{17} - \frac{21i}{17} = \frac{31}{17} - i\frac{21}{17} \]

Plot the number in the complex plane

Plot the complex number \(\frac{31}{17} - i\frac{21}{17}\) in the complex plane. The real part is \(\frac{31}{17}\) and the imaginary part is \(-\frac{21}{17}\).

Key concepts

Complex Plane

When we talk about the complex plane, we're referring to a way to visualize complex numbers. This plane is a two-dimensional space where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.
The form of a complex number is generally written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
To plot \( \frac{31}{17} - i\frac{21}{17} \) on the complex plane, follow these steps:
- Find the real part, \( \frac{31}{17} \), and plot it along the horizontal axis.
- Find the imaginary part, \( - \frac{21}{17} \), and plot it along the vertical axis.
- Combine these points to place the complex number on the plane.
This helps us visualize complex numbers in the same way we do regular numbers on a number line.

Rationalizing Denominator

Rationalizing the denominator means to eliminate any complex numbers from the denominator of a fraction.
We achieve this by multiplying both the numerator and the denominator by the complex conjugate of the denominator.
In our exercise, the problem \( \frac{3i-7}{i+4} \) requires us to rationalize the denominator. The denominator here is \(i+4\), and its complex conjugate is \(-i+4\).
By multiplying both the numerator and the denominator by \(-i+4\), we transform the denominator into a real number, thereby simplifying the fraction to a standard form.

Complex Conjugate

The complex conjugate of a complex number \(a+bi\) is \(a-bi\).
The conjugate essentially changes the sign of the imaginary part but keeps the real part the same.
This process is vital when rationalizing the denominator.
In our sample problem, the complex number in the denominator is \(i+4\), and its conjugate is \(-i+4\).
Multiplying by the conjugate helps eliminate the imaginary parts in the denominators, simplifying the math involved.

Difference of Squares

The difference of squares is a mathematical formula that states \((a-b)(a+b) = a^2 - b^2\).
This formula is especially useful when dealing with complex numbers.
In the denominator of our problem, we have \((i+4)(-i+4)\).
Using the difference of squares, this becomes \( i^2 - 4^2 = -1 - 16 = -17 \).
This simplification helps us transform the denominator into a real number, making the fraction easier to work with.
Understanding this concept helps in rationalizing complex fractions and simplifying them properly.