(1+5i)⋅(−3−i)

2 min read Jun 16, 2024
(1+5i)⋅(−3−i)

Multiplying Complex Numbers: (1 + 5i) ⋅ (-3 - i)

This article explores the process of multiplying two complex numbers: (1 + 5i) ⋅ (-3 - i).

Understanding Complex Numbers

Complex numbers are numbers 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).

Multiplication Process

To multiply complex numbers, we use the distributive property (also known as FOIL method) similar to multiplying binomials:

  1. Multiply the first terms: (1) ⋅ (-3) = -3
  2. Multiply the outer terms: (1) ⋅ (-i) = -i
  3. Multiply the inner terms: (5i) ⋅ (-3) = -15i
  4. Multiply the last terms: (5i) ⋅ (-i) = -5i²

Now, we have: -3 - i - 15i - 5i²

Since i² = -1, we can substitute it: -3 - i - 15i - 5(-1)

Simplifying the expression: -3 - i - 15i + 5

Combining real and imaginary terms: (-3 + 5) + (-1 - 15)i

Therefore, the product of (1 + 5i) ⋅ (-3 - i) is 2 - 16i.

Conclusion

Multiplying complex numbers involves using the distributive property and simplifying the expression by substituting i² with -1. This process results in a new complex number, in this case, 2 - 16i.

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