Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$(1.6+3.2 i)+(-5.8+4.3 i)$$

Step-by-Step Solution

Answer

The result of the addition is \(-4.2 + 7.5i\).

Step 1: Identify the real and the imaginary parts

In this case, the real parts are \(1.6\) and \(-5.8\) and the imaginary parts are \(3.2i\) and \(4.3i\).

Step 2: Add the real parts

Adding \(1.6\) and \(-5.8\) gives \(-4.2\). So, the real part of the result is \(-4.2\).

Step 3: Add the imaginary parts

Adding \(3.2i\) and \(4.3i\) gives \(7.5i\). So, the imaginary part of the result is \(7.5i\).

Step 4: Write the result in standard form

The resulting complex number, in standard form, is \(-4.2 + 7.5i\). This is the sum of the two given complex numbers.

Key Concepts

Addition of Complex Numbers

Adding complex numbers is like a two-step dance. First, you add the real parts, and then you add the imaginary parts. Let's break this down.

• Real part addition - Consider it like adding plain old numbers. In our example given in the exercise, we add the real component of each number: \(1.6 + (-5.8) = -4.2\).
• Imaginary part addition - This is where we take the imaginary pieces, ignoring the 'i' for now, and add them: \(3.2 + 4.3 = 7.5\). Then, we just tack on the 'i', getting \(7.5i\).

The key is to treat the real and imaginary parts separately during addition. Just remember, real with real, imaginary with imaginary.

Standard Form of a Complex Number

The standard form of a complex number is essentially how complex numbers are presented. It's the format that makes them easier to work with and understand.
A complex number in standard form is written as \(a + bi\), where:

• \(a\) is the real part, which is an ordinary real number.
• \(bi\) is the imaginary part, with \(b\) being a real number coefficient and \(i\) representing the square root of \(-1\).

For the given exercise, after adding the real and imaginary parts, our result was \(-4.2 + 7.5i\). This shows the standard form with \(-4.2\) as the real part and \(7.5i\) as the imaginary part. So, whenever you see a complex number, remember its friend—the standard form.

Imaginary and Real Parts

Every complex number is a combination of two parts: a real part and an imaginary part. Recognizing these parts is crucial in tackling problems with complex numbers.

• Real Part : This is the same as the regular numbers you use daily. It is the "a" in \(a + bi\).
• Imaginary Part : This is the bit that involves the letter \(i\), which represents the square root of \(-1\). It's \(bi\) in the standard form. The imaginary part holds a "b" value (just a real number) multiplied by \(i\).

In the exercise, the real parts are \(1.6\) and \(-5.8\) while the imaginary parts are \(3.2i\) and \(4.3i\). These help us organize and simplify the process of arithmetic with complex numbers. By identifying them, we can precisely execute operations like addition.