(1-3i)(1+3i)

2 min read Jun 16, 2024
(1-3i)(1+3i)

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):

  1. First: Multiply the first terms of each binomial: 1 * 1 = 1
  2. Outer: Multiply the outer terms of each binomial: 1 * 3i = 3i
  3. Inner: Multiply the inner terms of each binomial: -3i * 1 = -3i
  4. 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.

Related Post


Featured Posts