Multiplying Complex Numbers: (1-3i)(1+3i)
This article will walk you through the multiplication of two complex numbers, (1 - 3i) and (1 + 3i), and explain the resulting outcome.
Understanding Complex Numbers
Complex numbers are numbers that include both a real and an imaginary component. They are expressed in the form a + bi, where:
- a is the real part
- b is the imaginary part
- i is the imaginary unit, where i² = -1
Multiplication Process
To multiply complex numbers, we use the distributive property (also known as the FOIL method):
- First: Multiply the first terms of each binomial: 1 * 1 = 1
- Outer: Multiply the outer terms of each binomial: 1 * 3i = 3i
- Inner: Multiply the inner terms of each binomial: -3i * 1 = -3i
- Last: Multiply the last terms of each binomial: -3i * 3i = -9i²
This gives us: 1 + 3i - 3i - 9i²
Simplifying the Result
Remember that i² = -1. Substituting this into our result, we get:
1 + 3i - 3i - 9(-1)
Simplifying further:
1 + 3i - 3i + 9
Combining the real and imaginary terms:
(1 + 9) + (3 - 3)i
Finally, we arrive at:
10 + 0i = 10
Conclusion
The product of (1 - 3i) and (1 + 3i) is 10. This result demonstrates an interesting property of complex numbers: when a complex number is multiplied by its conjugate (obtained by changing the sign of the imaginary part), the result is always a real number.