Simplify Complex Number: (-9 + 9i) / (-3 + 5i) Solution
Complex numbers might seem intimidating at first, but with a systematic approach, you can easily simplify even the trickiest expressions. In this article, we'll break down the process of simplifying the complex number expression (-9 + 9i) / (-3 + 5i) step by step. So, grab your pen and paper, and let's dive into the world of complex numbers!
Understanding Complex Numbers
Before we jump into the simplification, let's quickly recap what complex numbers are. 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 the square root of -1 (i.e., i² = -1). The a part is called the real part, and the b part is called the imaginary part.
When dealing with division involving complex numbers, our goal is to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. But what is a conjugate? The conjugate of a complex number a + bi is a - bi. We simply change the sign of the imaginary part.
Step 1: Identify the Complex Numbers
In our expression (-9 + 9i) / (-3 + 5i), we have two complex numbers:
- Numerator: -9 + 9i
- Denominator: -3 + 5i
Our task is to simplify this fraction by removing the imaginary component from the denominator.
Step 2: Find the Conjugate of the Denominator
The denominator is -3 + 5i. To find its conjugate, we change the sign of the imaginary part. Therefore, the conjugate of -3 + 5i is -3 - 5i.
This is a crucial step. The conjugate allows us to eliminate the imaginary part from the denominator because when a complex number is multiplied by its conjugate, the result is always a real number. This is because the imaginary terms cancel each other out due to the difference of squares pattern: (a + bi)(a - bi) = a² - (bi)² = a² + b².
Step 3: Multiply Numerator and Denominator by the Conjugate
Now, we multiply both the numerator and the denominator of our original expression by the conjugate we just found (-3 - 5i). This is similar to multiplying a fraction by 1 (in the form of (conjugate)/(conjugate)), which doesn't change its value but transforms its appearance.
So, we have:
((-9 + 9i) / (-3 + 5i)) * ((-3 - 5i) / (-3 - 5i))
This might look complicated, but don't worry, we'll break it down further.
Step 4: Expand the Numerator and Denominator
Next, we need to expand both the numerator and the denominator using the distributive property (also known as the FOIL method):
Numerator:
(-9 + 9i) * (-3 - 5i) = (-9 * -3) + (-9 * -5i) + (9i * -3) + (9i * -5i)
Simplifying this gives us:
27 + 45i - 27i - 45i²
Denominator:
(-3 + 5i) * (-3 - 5i) = (-3 * -3) + (-3 * -5i) + (5i * -3) + (5i * -5i)
Simplifying this gives us:
9 + 15i - 15i - 25i²
Notice how the middle terms (+15i and -15i) in the denominator cancel each other out. This is exactly what we wanted to achieve by multiplying by the conjugate!
Step 5: Simplify Using i² = -1
Remember that i² is equal to -1. We can now substitute -1 for i² in both the numerator and the denominator.
Numerator:
27 + 45i - 27i - 45(-1) = 27 + 45i - 27i + 45
Denominator:
9 + 15i - 15i - 25(-1) = 9 + 25
Step 6: Combine Like Terms
Now, let's combine the real and imaginary terms separately in both the numerator and the denominator.
Numerator:
(27 + 45) + (45i - 27i) = 72 + 18i
Denominator:
9 + 25 = 34
So, our expression now looks like this:
(72 + 18i) / 34
Step 7: Separate into Real and Imaginary Parts
To express the complex number in its standard form (a + bi), we can separate the fraction into two parts:
(72 / 34) + (18i / 34)
Step 8: Simplify the Fractions
Finally, we simplify both fractions by finding the greatest common divisor (GCD) for the numerator and denominator.
For 72/34, the GCD is 2. Dividing both by 2, we get 36/17.
For 18/34, the GCD is 2. Dividing both by 2, we get 9/17.
The Simplified Complex Number
Therefore, the simplified form of the complex number expression (-9 + 9i) / (-3 + 5i) is:
(36/17) + (9/17)i
This is the complex number expressed in its standard form, with a real part of 36/17 and an imaginary part of 9/17.
Conclusion
Simplifying complex numbers might seem challenging at first, but by following these steps – identifying the complex numbers, finding the conjugate, multiplying, expanding, simplifying using i² = -1, combining like terms, and separating into real and imaginary parts – you can confidently tackle any complex number division problem. Remember, the key is to eliminate the imaginary part from the denominator by using the conjugate. Keep practicing, and you'll become a pro at simplifying complex numbers in no time!
For further reading and more examples on complex numbers, you can explore resources like Khan Academy's Complex Numbers section.
