Complex Number Calculation: Real, Imaginary, and Absolute Value Explanation

In summary, complex number calculation involves using both real and imaginary numbers to represent quantities in a two-dimensional plane. Real numbers are represented on the horizontal axis, while imaginary numbers are represented on the vertical axis. The absolute value of a complex number is the distance from the origin to its point on the plane, and is calculated using the Pythagorean Theorem. This concept is essential in understanding the behavior of complex numbers and their applications in mathematics and science.
  • #1
JaysFan31
Calculate the real part, the imaginary part, and the absolute value of the following expression:

i * [(1+2i)(5-3i)+3i/(1+i)].


So I did the math out this way:

(1+2i)(5-3i)= 11+7i
(11+7i)+3i/(1+i)= (4+21i)/(1+i)
i * [(4+21i)/(1+i)] = (4i-21)/(1+i)

Is this correct and what do you call the imaginary part and the real part if a denominator exists with an imaginary i?

Thanks for any help.
 
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  • #2
Yeah I figured it out. You just multiply it by its conjugate earlier in the process.
 

1. What are complex numbers?

Complex numbers are numbers that contain both a real part and an imaginary part. They are often represented in the form a + bi, where a is the real part and bi is the imaginary part, with i denoting the imaginary unit (the square root of -1).

2. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For example, (3 + 2i) + (5 - 4i) = (3 + 5) + (2i - 4i) = 8 - 2i. Similarly, (3 + 2i) - (5 - 4i) = (3 - 5) + (2i + 4i) = -2 + 6i.

3. What about multiplying and dividing complex numbers?

Multiplication and division of complex numbers is done in a similar way as with real numbers, but using the rules for multiplying and dividing complex numbers. For multiplication, you multiply the real parts and the imaginary parts separately and then combine them. For division, you can use the conjugate of the denominator to simplify the expression.

4. What is the absolute value of a complex number?

The absolute value (also known as modulus or magnitude) of a complex number is the distance between the number and the origin (0,0) on the complex plane. It is calculated by taking the square root of the sum of the squares of the real and imaginary parts. In other words, the absolute value of a complex number a + bi is |a + bi| = √(a^2 + b^2).

5. How are complex numbers used in real life?

Complex numbers have many applications in mathematics, physics, engineering, and other fields. They are used to represent and solve problems involving alternating currents, oscillations, and electromagnetic fields. They are also used in signal processing, control systems, and computer graphics. In everyday life, complex numbers are commonly used in electrical engineering, telecommunications, and digital signal processing.

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