Problem 16 - Contemporary Precalculus

Question

If \(z=3-4 i,\) find \(|z|^{2}\) and \(z \bar{z},\) where \(\bar{z}\) is the conjugate of \(z\) (see page 323 ).

Step-by-Step Solution

Answer

Question: Find the modulus squared and the product of the complex number and its conjugate for the complex number \(z = 3 - 4i\). Answer: The modulus squared of the complex number \(z\) is \(|z|^2 = 25\), and the product of the complex number and its conjugate is \(z \bar{z} = 25\).

Step 1: Find the conjugate of the complex number

To find the conjugate of a complex number, we simply change the sign of the imaginary part. In this case, \(z = 3 - 4i\), so the conjugate is \(\bar{z} = 3 + 4i\).

Step 2: Calculate the modulus squared of the complex number

To calculate the modulus squared of a complex number, we find the sum of the squares of the real part and the imaginary part. For \(z = 3 - 4i\), we have \(|z|^2 = 3^2 + (-4)^2 = 9 + 16 = 25\).

Step 3: Calculate the product of the complex number and its conjugate

To calculate \(z \bar{z}\), we multiply the complex number \(z\) by its conjugate \(\bar{z}\). Using the definition of complex number multiplication, we have: \(z \bar{z} = (3 - 4i)(3 + 4i) = 3(3) + 3(4i) - 4i(3) - 4i(4i) = 9 + 12i - 12i - 16i^2\). Since \(i^2 = -1\), we can simplify the expression to: \(z \bar{z} = 9 - 16(-1) = 9 + 16 = 25\). Thus, the modulus squared of \(z\) is \(|z|^2 = 25\) and the product of \(z\) and its conjugate is \(z \bar{z} = 25\).

Key Concepts

Conjugate of a Complex Number

In the realm of complex numbers, the concept of a conjugate is a fundamental one. To obtain the conjugate of a complex number, you simply switch the sign of the imaginary component. If we have a complex number in the form of \(z = a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, the conjugate \(\bar{z}\) will be \(a - bi\). It's like looking in a mirror where only the sign of the imaginary part is changed.

For instance, the complex number \(z = 3 - 4i\) has a counterpart, its conjugate \(\bar{z} = 3 + 4i\). It's a simple operation, yet it holds significant importance, particularly when it comes to complex number multiplication and finding the modulus of a complex number, as it comes into play in both situations.

Modulus of a Complex Number

The modulus of a complex number, often represented by \(|z|\), is a measure of its size or magnitude. Think of it as the distance of the number from the origin in a two-dimensional plane where the x-axis represents real numbers and the y-axis stands for imaginary numbers. To find this distance, you use the Pythagorean theorem, squaring the real part and the imaginary part, then taking the square root of their sum.

Given our number \(z = 3 - 4i\), calculating \(|z|^2\) involves squaring both 3 (the real part) and -4 (the imaginary part), yielding \(9 + 16\), which equates to 25. Therefore, the modulus \(|z|\) itself would be the square root of 25, which is 5. This squared modulus is especially useful as it appears frequently in both mathematical theory and applied physical problems.

Complex Number Multiplication

Multiplying complex numbers can be likened to multiplying binomials, where one must pay attention to the special property that \(i^2 = -1\). To multiply a complex number by its conjugate, like \(z\) times \(\bar{z}\), you apply the distributive property, multiply each part accordingly, and simplify using the imaginary unit's property.

For the number \(z = 3 - 4i\), multiplied by its conjugate \(\bar{z} = 3 + 4i\), you would expand the product to \(9 + 12i - 12i - 16i^2\). Because the imaginary parts (\(12i\) and \(-12i\)) cancel each other out and \(i^2\) is substituted by \(-1\), you are left with \(9 + 16\), which simplifies to 25. This multiplication reveals an essential characteristic: a complex number multiplied by its conjugate always results in a non-negative real number, which is the squared modulus of the original complex number.