(2%2B3i)(3%2B4i).jpeg "(1+2i)(2+3i)(3+4i)")
Multiplying Complex Numbers: (1+2i)(2+3i)(3+4i)
This article explores the multiplication of complex numbers, specifically the expression: (1+2i)(2+3i)(3+4i).
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.
Multiplying Complex Numbers
To multiply complex numbers, we distribute just as we would with regular binomials.
For example:
(1+2i)(2+3i) = 1(2) + 1(3i) + 2i(2) + 2i(3i) = 2 + 3i + 4i + 6i² = 2 + 7i - 6 = -4 + 7i
Calculating (1+2i)(2+3i)(3+4i)
To calculate the given expression, we perform the multiplication in steps:
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Multiply (1+2i)(2+3i): As shown above, (1+2i)(2+3i) = -4 + 7i
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Multiply the result by (3+4i): (-4 + 7i)(3 + 4i) = -4(3) - 4(4i) + 7i(3) + 7i(4i) = -12 - 16i + 21i + 28i² = -12 + 5i - 28 = -40 + 5i
Therefore, (1+2i)(2+3i)(3+4i) = -40 + 5i.
Summary
Multiplying complex numbers involves distributing and simplifying the expression. Remember that i² = -1, and always combine the real and imaginary parts of the final result.