91Ó°ÊÓ

Complex numbers are a combination of real and imaginary numbers. They take the form of \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part. Here, \(i\) is the imaginary unit, satisfying \(i^2 = -1\). This concept extends the idea of numbers to include solutions to equations that don't have real solutions. For example, the equation \(x^2 + 1 = 0\) has no real solution, but it has two complex solutions, \(i\) and \(-i\). Understanding complex numbers is crucial for solving many mathematical problems, especially those involving quadratic equations and characteristic equations in differential equations.

Subtracting complex numbers involves subtracting both the real parts and the imaginary parts separately. Let's look at the given problem: \((5 + \frac{1}{3}i) - (\frac{1}{2} - \frac{1}{2}i)\). To perform the subtraction, follow these steps:
1. Distribute the negative sign to both terms inside the parentheses. This will give you: \(5 + \frac{1}{3}i - \frac{1}{2} + \frac{1}{2}i\).
2. Combine the real parts: \(5 - \frac{1}{2}\).
3. Combine the imaginary parts: \(\frac{1}{3}i + \frac{1}{2}i\).
4. Simplify the real parts: \(5 - \frac{1}{2} = \frac{10}{2} - \frac{1}{2} = \frac{9}{2}\).
5. Simplify the imaginary parts: \(\frac{1}{3}i + \frac{1}{2}i = \frac{2}{6}i + \frac{3}{6}i = \frac{5}{6}i\).
6. Combine the simplified parts together to get the final result: \(\frac{9}{2} + \frac{5}{6}i\).

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). This means \(i^2 = -1\). An imaginary number takes the form \(bi\), where \(b\) is a real number. Complex numbers use imaginary numbers to form their imaginary part. For example, in the complex number \(5 + \frac{1}{3}i\), \(\frac{1}{3}i\) is the imaginary part. Imaginary numbers are used in many areas of engineering and physics, including signal processing and quantum mechanics.

Real numbers play a key role in complex number operations. In arithmetic involving complex numbers, the real parts and imaginary parts are treated separately. For instance, when performing the subtraction \((5 + \frac{1}{3}i) - (\frac{1}{2} - \frac{1}{2}i)\), we handle the real numbers (5 and \(\frac{1}{2}\)) and the coefficients of the imaginary part (\(\frac{1}{3}\) and \(\frac{1}{2}\)) independently. To simplify the operation:
1. Subtract the real parts: \(5 - \frac{1}{2} = \frac{9}{2}\).
2. Combine the imaginary parts: \(\frac{1}{3}i + \frac{1}{2}i\).
Thus, complex number problems often involve basic arithmetic with real numbers, making it fundamental to have a strong understanding of real number operations to handle complex arithmetic confidently.