(5-3i)-in-standard-form.jpeg "(5+3i)(5-3i) In Standard Form")
Simplifying Complex Numbers: (5 + 3i)(5 - 3i)
In mathematics, complex numbers are 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.
This article will walk through simplifying the expression (5 + 3i)(5 - 3i) into standard form.
Understanding Complex Numbers
- Real Part: The real part of a complex number is the term without the imaginary unit i. In the expression (5 + 3i), the real part is 5.
- Imaginary Part: The imaginary part of a complex number is the term multiplied by the imaginary unit i. In the expression (5 + 3i), the imaginary part is 3.
Simplifying the Expression
We can simplify the expression (5 + 3i)(5 - 3i) by applying the distributive property (also known as FOIL).
1. Distribute the terms: (5 + 3i)(5 - 3i) = (5 * 5) + (5 * -3i) + (3i * 5) + (3i * -3i)
2. Simplify the multiplication: = 25 - 15i + 15i - 9i²
3. Remember that i² = -1: = 25 - 15i + 15i - 9(-1)
4. Combine like terms: = 25 + 9
5. Final Answer: = 34
Conclusion
By applying the distributive property and substituting i² with -1, we have successfully simplified the expression (5 + 3i)(5 - 3i) to its standard form, 34. This result demonstrates an important property of complex numbers: multiplying a complex number by its conjugate (the complex number with the opposite sign for the imaginary part) results in a real number. In this case, the conjugate of (5 + 3i) is (5 - 3i).