Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$\left(\frac{3}{2}+\frac{5}{2} i\right)+\left(\frac{5}{3}+\frac{11}{3} i\right)$$

Step-by-Step Solution

Answer

The sum of the complex numbers \(\frac{3}{2}+\frac{5}{2}i\) and \(\frac{5}{3}+\frac{11}{3}i\) is \(\frac{19}{6} + \frac{17}{3}i\).

Step 1: Identify Real and Imaginary Parts

In the given complex numbers \(\frac{3}{2}+\frac{5}{2}i\) and \(\frac{5}{3}+\frac{11}{3}i\), the real parts are \(\frac{3}{2}\) and \(\frac{5}{3}\) while the imaginary parts are \(\frac{5}{2}\) and \(\frac{11}{3}\) respectively.

Step 2: Add Real Parts

Add the real parts \(\frac{3}{2}\) and \(\frac{5}{3}\) together to get the total real part.

Step 3: Add Imaginary Parts

Add the imaginary parts \(\frac{5}{2}\) and \(\frac{11}{3}\) together to get the total imaginary part.

Step 4: Express in Standard Form

The final step is to express the answer in standard form, which is a + bi where a is the real part and b is the imaginary part.

Key Concepts

Standard Form of Complex Numbers

Complex numbers are an essential part of mathematics, extending the idea of the traditional number system to include the square root of negative one, typically represented as the imaginary unit 'i'. Understanding the standard form of a complex number is crucial because it simplifies the process of performing algebraic operations. The standard form is expressed as 'a + bi', where 'a' is known as the real part, and 'bi' is the imaginary part, with 'i' representing the square root of -1.

The real part is a real number, meaning it lies along the number line that we are all familiar with. The imaginary part, on the other hand, involves a real number multiplied by the imaginary unit 'i'. It's important not only to identify these parts but also to know how to work with them in calculations, as seen in the exercise involving the addition of two complex numbers in standard form.

Real and Imaginary Parts

In the realm of complex numbers, identifying the real and imaginary parts is a fundamental skill. Being able to distinguish between these two components is a prerequisite for any algebraic operation with complex numbers. In the original exercise, we have two complex numbers \( \frac{3}{2}+\frac{5}{2}i \) and \( \frac{5}{3}+\frac{11}{3}i \) where \( \frac{3}{2} \) and \( \frac{5}{3} \) are the real parts, and \( \frac{5}{2}i \) and \( \frac{11}{3}i \) are the imaginary parts.

It's worth noting that the real part can be any real number, including zero, while the imaginary part will always have 'i' attached. This is why when adding complex numbers, as in the exercise, we combine real parts with real parts and imaginary parts with imaginary parts, treating them almost as like terms in algebra.

Algebraic Operations with Complex Numbers

Algebraic operations with complex numbers follow the same foundational principles as those with real numbers, with the addition of rules concerning the imaginary unit 'i'. When adding complex numbers, we perform the addition separately for the real parts and the imaginary parts. This method was demonstrated in the provided exercise, where the addition is split into two parts.

• For the real parts: \( \frac{3}{2} + \frac{5}{3} \).
• For the imaginary parts: \( \frac{5}{2}i + \frac{11}{3}i \) (we factor out the 'i' and add the coefficients as real numbers).

Once we have the sums of the real and imaginary parts, we can write them together in standard form to express the result of the addition. The beauty of complex numbers lies in the predictability and structure of such operations, which can be performed even with multiplication, division, and subtraction of complex numbers, always adhering to the rules for 'i' and combining like parts.