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(2+3i)(1-4i)

2 min read Jun 16, 2024
(2+3i)(1-4i)

Multiplying Complex Numbers: (2 + 3i)(1 - 4i)

This article will walk through the process of multiplying the complex numbers (2 + 3i) and (1 - 4i).

Understanding Complex Numbers

Before we begin, let's quickly recap what complex numbers are. 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).

Multiplication Process

To multiply complex numbers, we use the distributive property, just like we would with any binomial multiplication.

  1. FOIL (First, Outer, Inner, Last):

    • First: 2 * 1 = 2
    • Outer: 2 * (-4i) = -8i
    • Inner: 3i * 1 = 3i
    • Last: 3i * (-4i) = -12i²

  2. Combine like terms:

    • 2 - 8i + 3i - 12i²

  3. Substitute i² = -1:

    • 2 - 8i + 3i - 12(-1)

  4. Simplify:

    • 2 - 8i + 3i + 12

  5. Combine real and imaginary parts:

    • (2 + 12) + (-8 + 3)i

  6. Final result:

    • 14 - 5i

Conclusion

Therefore, the product of (2 + 3i) and (1 - 4i) is 14 - 5i.

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