(9-5i)-in-standard-form.jpeg "(9+5i)(9-5i) In Standard Form")
Simplifying Complex Numbers: (9 + 5i)(9 - 5i)
This article explores the simplification of the complex number expression (9 + 5i)(9 - 5i) into its standard form (a + bi).
Understanding Complex Numbers
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 (i² = -1).
Simplifying the Expression
The expression (9 + 5i)(9 - 5i) represents the product of two complex numbers. To simplify this expression, we can use the difference of squares pattern:
(a + b)(a - b) = a² - b²
Applying this pattern to our complex numbers:
(9 + 5i)(9 - 5i) = 9² - (5i)²
Now, we can simplify further by remembering that i² = -1:
9² - (5i)² = 81 - 25(-1) = 81 + 25
Finally, combining the real terms, we get:
81 + 25 = 106
Conclusion
Therefore, the simplified standard form of the expression (9 + 5i)(9 - 5i) is 106. This demonstrates that the product of a complex number and its conjugate (the same number with the opposite sign on the imaginary part) results in a real number.