In the preceding sections, we have primarily dealt with real numbers like 2, 13, 3, and 7.09. However, in this section, we will introduce a new concept called imaginary numbers. These numbers may seem elusive at first glance, but understanding them will expand our understanding of numbers and their applications.

To grasp this concept, let us consider the square root of 2. As we know, this yields a non-repeating decimal value of 1.414213562... But what about the square root of -2? You may assume that there is no solution to the square root of a negative number, but that is where imaginary numbers come into play.

The core idea of imaginary numbers is rooted in the imaginary unit, represented by the letter i. This unit is derived from the equation i^{2} = -1, which can also be expressed as -1 = i. Therefore, the square root of -2 can be written as -1 × 2 = -12 = i^{2} = 1.414i (correct to 3 decimal places). This imaginary unit is the key to understanding and working with imaginary numbers.

A complex number is an algebraic expression that includes the imaginary unit i = √-1 and is written in the form z = a + bi. It has two parts: a real part (a) and an imaginary part (bi). These numbers can also be denoted as z = Re(z) + Im(z)i = a+bi.

There are two important subcategories of complex numbers: real and imaginary. If the imaginary part (b) is equal to 0, then the complex number is considered to be a real number. On the other hand, if the real part (a) is equal to 0, it is said to be purely imaginary.

The applications of complex numbers are vast, ranging from fields like electrical engineering to quantum mechanics. They also help us solve polynomial equations that have no real solutions, as demonstrated in our previous article "Graph and Solve Quadratic Equations". Complex numbers provide a powerful tool for solving complex problems in various disciplines.

Complex numbers can be added, subtracted, multiplied, and divided following certain rules. Let's explore these operations in detail.

Adding or subtracting complex numbers is a straightforward process. Simply add or subtract their corresponding real and imaginary parts. For example, let z1 and z2 be two complex numbers with z1 = a + bi and z2 = c + di. Their sum would be z1 + z2 = (a + c) + (b + d)i, and their difference would be z1 - z2 = (a - c) + (b - d)i.

Scalar multiplication of complex numbers involves multiplying a real number (or scalar) with a complex number. To do this, we simply multiply both the real and imaginary parts of the complex number by the scalar. For instance, if z = a + bi and c is the scalar, then c×z = c(a + bi) = ca + cbi.

Multiplying complex numbers follows the same principles as the binomial expansion technique. We apply the FOIL method and combine like terms. In this case, let z1 and z2 be two complex numbers with z1 = a + bi and z2 = c + di. The multiplication would involve the following steps:

- Write both numbers in standard form.
- Perform the binomial expansion.
- Combine like terms.

In summary, complex numbers are a crucial concept in mathematics with practical applications in various fields. They consist of real and imaginary numbers and can be added, subtracted, multiplied, and divided following specific rules. Through complex numbers, we can solve complex problems and expand our understanding of numbers.

Complex numbers involve both real and imaginary components, and performing arithmetic operations with them requires some knowledge of algebra. Let's explore the FOIL method and a specific formula for multiplication and division of complex numbers.

Multiplying two complex numbers involves multiplying their real and imaginary parts. For instance, z_{1} = a + bi and z_{2} = c + di, their product would be z_{1} × z_{2} = (ac - bd) + (ad + bc)i. This is similar to the FOIL method from algebra, with the real part coming from multiplying the real parts and the imaginary part from multiplying the imaginary parts.

Dividing complex numbers follows a specific formula. Let z_{1} = a + bi and z_{2} = c + di and to divide z_{1} by z_{2}, we first multiply the numerator and denominator by the complex conjugate of z_{2}, which is z_{2}* = c - di. After simplification, the answer is written in standard form a + bi, such as (29 + 11i)/74 for α = 3 - 2i and β = 5 + 7i.

- Addition: z
_{1}+ z_{2}= (a + c) + (b + d)i - Subtraction: z
_{1}- z_{2}= (a - c) + (b - d)i - Multiplication: z
_{1}× z_{2}= (ac - bd) + (ad + bc)i - Division: z
_{1}/z_{2}= (ac + bd) + (bc - ad)i / (c^{2}+ d^{2})

Operations with complex numbers involve identifying and manipulating their real and imaginary parts. Remember to always write the final answer in standard form a + bi for clarity. By understanding these fundamental concepts, you can confidently solve complex number problems. Happy calculating!

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