(1+i)⋅(3−5i)

Jasonbradley

2 min read

·

Jun 16, 2024

12

Share

Complex Number Multiplication: (1+i)⋅(3-5i)

This article will explore the multiplication of the complex numbers (1+i) and (3-5i).

Understanding Complex Numbers

Complex numbers are numbers that extend the real number system by including the imaginary unit 'i', where i² = -1. They are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.

Multiplication of Complex Numbers

To multiply complex numbers, we use the distributive property, just as we do with real numbers. We multiply each term in the first complex number by each term in the second complex number.

(1+i)⋅(3-5i) Calculation

1. Expand using the distributive property:

   (1+i)⋅(3-5i) = (1 * 3) + (1 * -5i) + (i * 3) + (i * -5i)

2. Simplify:

   = 3 - 5i + 3i - 5i²

3. Substitute i² = -1:

   = 3 - 5i + 3i - 5(-1)

4. Combine real and imaginary terms:

   = (3 + 5) + (-5 + 3)i

5. Final result:

   = 8 - 2i

Conclusion

Therefore, the product of (1+i) and (3-5i) is 8 - 2i. This process demonstrates how complex numbers can be multiplied using the distributive property and the understanding that i² = -1.