Problem 8 - Precalculus

Question

If \(z=1+4 i,\) compute \(z-10 i\)

Step-by-Step Solution

Answer

The result is \(1 - 6i\).

Step 1: Identify the given complex number

We are given the complex number \(z = 1 + 4i\). This number is composed of a real part and an imaginary part.

Step 2: Identify the operation we need to perform

The problem asks us to compute \(z - 10i\). This requires us to subtract the imaginary part \(10i\) from the complex number \(z\).

Step 3: Perform the subtraction on the imaginary parts

Subtract the imaginary part: \((4i - 10i)\). This results in \(-6i\).

Step 4: Write the resulting complex number

The real part of \(z\) remains unchanged, so the result is \(1 - 6i\). Therefore, \(z - 10i = 1 - 6i\).

Key Concepts

Imaginary Part

A complex number such as \( z = 1 + 4i \) comprises two components: a real part and an imaginary part. The imaginary part is the component connected to the imaginary unit \( i \). The imaginary unit is a mathematical concept that facilitates calculations involving numbers that aren't real. It satisfies the equation \( i^2 = -1 \).

In this example, \( 4i \) is the imaginary part of the complex number \( 1 + 4i \). Often, students contend with the idea of what the imaginary part represents since it doesn't correlate directly to our number system's traditional quantities. Imaginary numbers, however, provide us with a broader set of tools for solving equations and understanding complex phenomena.

• Imaginary part of \( 1 + 4i \): \( 4i \)
• Always includes the variable \( i \), which represents the square root of -1
• Multiplied by a real number, giving the full imaginary component

Real Part

In a complex number such as \( z = 1 + 4i \), the real part is the component without the imaginary unit \( i \). It is simply a regular real number.

In our exercise, the real part of the complex number \( 1 + 4i \) is \( 1 \). This portion of the complex number is akin to numbers we encounter in daily life, without any imaginary component involved. Even during operation like subtraction with another imaginary number, the real part remains unchanged unless we are directly modifying or combining real numbers.

• Real part of \( 1 + 4i \): \( 1 \)
• Maintains its value when only the imaginary part is altered
• Behaves like standard real numbers in arithmetic operations

Complex Number Subtraction

Subtracting complex numbers involves the subtraction of their respective real parts and imaginary parts. If only the imaginary part is being subtracted, the real part of the original complex number remains unaffected.

In the exercise, we are tasked with calculating \( z - 10i \). Given \( z = 1 + 4i \), we subtract \( 10i \) from the imaginary portion. The calculation \( 4i - 10i \) yields \( -6i \). The real part, which is \( 1 \), stays the same.

Thus, the solution for \( z - 10i \) equals \( 1 - 6i \). This subtraction illustrates:

• A change only affecting the imaginary part: \( 4i - 10i = -6i \)
• No alteration to the real part: remains \( 1 \)
• Resulting complex number: \( 1 - 6i \)

Understanding subtraction in this way helps in grasping larger concepts in mathematics and seeing how changes to one component of a complex number affect the whole entity.