%2F(5-3i).jpeg "(3+7i)/(5-3i)")
Simplifying Complex Fractions: (3 + 7i) / (5 - 3i)
In mathematics, particularly in the realm of complex numbers, it's often necessary to simplify expressions involving complex fractions. This article will guide you through the process of simplifying the expression (3 + 7i) / (5 - 3i).
Understanding Complex Numbers
Before diving into the simplification, let's briefly 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
- i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1)
The Process of Simplification
The key to simplifying complex fractions is to eliminate the imaginary term in the denominator. We achieve this by multiplying both the numerator and denominator by the conjugate of the denominator.
1. Identify the Conjugate:
The conjugate of a complex number a + bi is a - bi. In our case, the conjugate of 5 - 3i is 5 + 3i.
2. Multiply by the Conjugate:
Multiply both the numerator and denominator of the fraction by the conjugate:
(3 + 7i) / (5 - 3i) * (5 + 3i) / (5 + 3i)
3. Expand and Simplify:
Expand both the numerator and denominator using the distributive property (FOIL method):
- Numerator: (3 + 7i)(5 + 3i) = 15 + 9i + 35i + 21i² = 15 + 44i - 21 = -6 + 44i
- Denominator: (5 - 3i)(5 + 3i) = 25 + 15i - 15i - 9i² = 25 + 9 = 34
4. Express the Result in Standard Form:
Now we have:
(-6 + 44i) / 34
Finally, separate the real and imaginary terms:
(-6/34) + (44/34)i
Simplifying the fractions, we get the final result:
(-3/17) + (22/17)i
Conclusion
By multiplying the fraction by the conjugate of the denominator, we successfully eliminated the imaginary term in the denominator and obtained the simplified complex number (-3/17) + (22/17)i. This process is essential for performing operations involving complex numbers and for expressing them in a standard form.