Understanding the Product of Complex Numbers: (2i)(3i)
In the world of mathematics, complex numbers play a crucial role in various fields, from engineering to quantum mechanics. When working with complex numbers, one of the fundamental operations is multiplication. In this article, we will delve into the details of finding the value of the product (2i)(3i).
What are Complex Numbers?
Complex numbers are a unique extension of the real number system, incorporating both a real part and an imaginary part. The imaginary unit, denoted as i, is defined as the square root of -1, or i² = -1. This imaginary unit allows us to represent and manipulate numbers that cannot be expressed solely using real numbers.
Multiplying Complex Numbers
The general rule for multiplying two complex numbers (a + bi) and (c + di) is:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
In the case of our specific problem, we have (2i)(3i). Let's break down the steps to find the value of this product.
- Identify the real and imaginary parts of the complex numbers:
(2i)has a real part of 0 and an imaginary part of 2.(3i)has a real part of 0 and an imaginary part of 3.
- Apply the multiplication rule:
(2i)(3i) = (0 × 0) - (2 × 3) + (0 × 3 + 2 × 0)i(2i)(3i) = -6 + 0i
Therefore, the value of the product (2i)(3i) is -6.
Understanding the Result
The result of -6 makes sense when we consider the properties of the imaginary unit i. Recall that i² = -1, which means that i × i = -1. In our case, we have (2i)(3i) = 2 × 3 × i × i = 6 × (-1) = -6.
This result demonstrates that the product of two imaginary numbers (in this case, 2i and 3i) is a real number, and the sign of the result is determined by the property i² = -1.
In summary, the value of the product (2i)(3i) is -6, which can be derived by applying the rules for multiplying complex numbers and understanding the properties of the imaginary unit i.