Complex Numbers Product Calculation

Understanding how to find the product of two complex numbers is a fundamental concept in mathematics. This problem asks us to multiply two specific complex numbers: \(1 + i\) and \(2 - 5i\). Let's break down the process step by step to find their product.

Complex Number Definition and Structure

A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\). The term \(a\) is called the real part, and \(b\) is called the imaginary part.

• The real part corresponds to the numbers on the standard number line.
• The imaginary part involves the imaginary unit \(i\), where \(i^2 = -1\). This property is crucial for simplifying expressions during complex number multiplication.

Product of Complex Numbers: Method Explained

To find the product of two complex numbers, say \( (a + bi) \) and \( (c + di) \), we use a method similar to multiplying two binomials in algebra, often referred to as the FOIL method (First, Outer, Inner, Last).

The steps involved are:

• Multiply the First terms of each binomial.
• Multiply the Outer terms.
• Multiply the Inner terms.
• Multiply the Last terms.
• Substitute \(i^2 = -1\) wherever it appears.
• Combine the real parts and the imaginary parts separately to express the final product in the standard \(a + bi\) form.

Product Calculation for (1 + i) and (2 - 5i)

Let's apply the method to find the product of the given complex numbers, \( (1 + i) \) and \( (2 - 5i) \).

Let \(z_1 = 1 + i\) and \(z_2 = 2 - 5i\).

We want to calculate \(z_1 \times z_2\):

\( (1 + i)(2 - 5i) \)

Now, let's perform the multiplication using the FOIL method:

1. First terms: \(1 \times 2 = 2\)
2. Outer terms: \(1 \times (-5i) = -5i\)
3. Inner terms: \(i \times 2 = 2i\)
4. Last terms: \(i \times (-5i) = -5i^2\)

Combining these results, we get:

\( 2 - 5i + 2i - 5i^2 \)

Next, we substitute the value of \(i^2\), which is \(-1\):

\( 2 - 5i + 2i - 5(-1) \)

\( 2 - 5i + 2i + 5 \)

Finally, we group and combine the real parts and the imaginary parts:

• Real parts: \(2 + 5 = 7\)
• Imaginary parts: \(-5i + 2i = -3i\)

So, the product of the two complex numbers is:

\( 7 - 3i \)

This is the final complex number product in the standard \(a + bi\) form.

Summary of Complex Number Multiplication

Term 1	Term 2	Product	Simplified Product
\(1\)	\(2\)	\(2\)	\(2\)
\(1\)	\(-5i\)	\(-5i\)	\(-5i\)
\(i\)	\(2\)	\(2i\)	\(2i\)
\(i\)	\(-5i\)	\(-5i^2\)	\(-5(-1) = 5\)
Total Product (Real + Imaginary)			\( (2 + 5) + (-5i + 2i) = 7 - 3i \)