Problem 2 - Algebra and Trigonometry

Question

For the complex number \(3+4 i\) the real part is _________ the imaginary part is ____________

Step-by-Step Solution

Answer

Real part: 3, Imaginary part: 4.

Step 1: Identify the Complex Number

We are given the complex number \(3+4i\). A complex number is of the form \(a + bi\) where \(a\) is the real part, and \(b\) is the imaginary part.

Step 2: Determine the Real Part

In the complex number \(3 + 4i\), the real part is the coefficient of \(1\) or the number before the \(i\). Here, the real part is \(3\).

Step 3: Determine the Imaginary Part

In the complex number \(3 + 4i\), the imaginary part is the coefficient of \(i\). Here, the imaginary part is \(4\).

Key Concepts

Real Part of a Complex Number

Complex numbers consist of two parts: the real part and the imaginary part. When dealing with complex numbers, it is essential to identify which part of the number is the real component. In a complex number of the form \(a + bi\), the real part is represented by \(a\).
\(a\) is simply the number without any imaginary unit \(i\). For example, in the complex number \(3 + 4i\), the real part is \(3\), because it appears in a position where no \(i\) follows it.
This component behaves like any ordinary real number that we are familiar with in basic arithmetic. Real parts can be positive, negative, or zero, and they are exactly what you would find on the number line.

Imaginary Part of a Complex Number

The imaginary part of a complex number is crucial for understanding these mathematical constructs. It is the part of the complex number that contains the imaginary unit \(i\), which denotes the square root of \(-1\). Imaginary parts are typically located after the real part in the expression of a complex number.
In the expression \(a + bi\), the imaginary part is \(b\). It is the coefficient of \(i\). For example, in the complex number \(3 + 4i\), the imaginary part is \(4\), because it is the number multiplying the \(i\). The imaginary part can also be positive, negative, or zero, much like the real part.

• Another example: in \(5 - 2i\), the imaginary part is \(-2\).

Understanding the imaginary part is fundamental for operations involving complex numbers.

Complex Number Form

Complex numbers have a standard format which is expressed as \(a + bi\). This form uniquely represents a complex number by combining both its real and imaginary parts.
In this format, \(a\) signifies the real component, while \(bi\) represents the imaginary component with \(b\) as its coefficient. Both parts together allow complex numbers to be utilized in advanced mathematical computations that real numbers alone cannot handle.

• Such applications include solving polynomial equations that aren't factorable using solely real numbers.
• They are used in electrical engineering to represent alternating currents.

The form \(a + bi\) facilitates addition, subtraction, multiplication, and division involving complex numbers by treating each part separately before recombining for the result.