Problem 98 - Calculus for Biology and Medicine

Question

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \overline{u z} $$

Step-by-Step Solution

Answer

The expression \(\overline{uz}\) simplifies to \(8-i\).

Step 1: Understand the Complex Conjugate

The complex conjugate of a complex number \(a+bi\) is \(a-bi\). For the given expression \(\overline{u z}\), we must first compute \(u z\) and then take its complex conjugate.

Step 2: Multiply the Complex Numbers

Multiply \(u=2-3i\) and \(z=1+2i\). Use the distributive property: \[(2-3i)(1+2i) = 2(1) + 2(2i) - 3i(1) - 3i(2i)\]Simplify the expression:\[= 2 + 4i - 3i - 6i^2\] Since \(i^2 = -1\), replace \(-6i^2\) with \(6\):\[= 2 + 4i - 3i + 6 = 8 + i\]

Step 3: Find the Complex Conjugate

Now that we have \(uz = 8+i\), compute its conjugate: \[\overline{8+i} = 8-i\]This means the complex conjugate of \(u z\) is \(8-i\).

Key Concepts

Complex Conjugate

The complex conjugate of a complex number is a very useful concept in mathematics. If you have a complex number written as \(a + bi\), its complex conjugate is \(a - bi\). Essentially, you just change the sign of the imaginary part. This is helpful in many calculations, particularly when dividing complex numbers, as it allows you to clear out the imaginary parts.

• The original complex number: \(a+bi\)
• The complex conjugate: \(a-bi\)

For instance, if you have the complex number \(8+i\), like in our exercise, its complex conjugate is simply \(8-i\). Knowing how to find the complex conjugate is key in working with complex equations and helps simplify the results of operations involving complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers might seem tricky at first, but it's quite similar to multiplying binomials. You apply the distributive property, also known as "foil" to some.

• Multiply the real part with the real part.
• Multiply the outer terms.
• Multiply the inner terms.
• Finally, multiply the imaginary parts together.

For example, in the exercise, we multiplied \(u = 2 - 3i\) by \(z = 1 + 2i\):\[(2 - 3i)(1 + 2i) = 2 \cdot 1 + 2 \cdot 2i - 3i \cdot 1 - 3i \cdot 2i\]Which resulted in:

• Real part: \(2 \cdot 1 = 2\)
• Imaginary part contributions: \(2 \cdot 2i - 3i = 4i - 3i\)
• Imaginary unit multiplication: \(-3i \cdot 2i = -6i^2\)

Since \(i^2\) is \(-1\), multiplying by \(-6i^2\) gives us \(6\). Adding all those up, we simplify to obtain \(8+i\). Remember to replace any \(i^2\) with \(-1\) to get the correct result.

Imaginary Unit

The imaginary unit, represented by \(i\), is a fundamental part of complex numbers. By definition, \(i\) is the square root of \(-1\), which is a somewhat abstract idea, but it forms the basis for the field of complex numbers.

• Imaginary unit: \(i\)
• Key property: \(i^2 = -1\)

This means any time \(i\) is squared, the result is \(-1\). This property is particularly crucial when we perform operations like multiplication with complex numbers. For example, in our solution, when we multiplied \(-3i\) and \(2i\), we used this property to simplify \(-6i^2\) to \(6\). Understanding this key aspect of \(i\) helps avoid mistakes in calculations and enables you to work comfortably with complex numbers and their various operations.