(1-4i).jpeg "(-3+2i)(1-4i)")
Multiplying Complex Numbers: (-3 + 2i)(1 - 4i)
This article will guide you through the process of multiplying two complex numbers: (-3 + 2i) and (1 - 4i).
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 (i.e., i² = -1).
Multiplication Process
To multiply complex numbers, we use the distributive property, similar to multiplying binomials in algebra.
Step 1: Expand the product
(-3 + 2i)(1 - 4i) = (-3)(1) + (-3)(-4i) + (2i)(1) + (2i)(-4i)
Step 2: Simplify using the distributive property
= -3 + 12i + 2i - 8i²
Step 3: Substitute i² with -1
= -3 + 12i + 2i - 8(-1)
Step 4: Combine real and imaginary terms
= (-3 + 8) + (12 + 2)i
Step 5: Simplify the result
= 5 + 14i
Conclusion
Therefore, the product of (-3 + 2i) and (1 - 4i) is 5 + 14i. Remember that when multiplying complex numbers, you essentially treat 'i' like any other variable, but keep in mind the key property that i² = -1.