(2%2B3i)(3%2B4i)(4%2B5i).jpeg "(1+i)(2+3i)(3+4i)(4+5i)")
Exploring the Product of Complex Numbers: (1+i)(2+3i)(3+4i)(4+5i)
This article will delve into the product of four complex numbers: (1+i)(2+3i)(3+4i)(4+5i). We will explore the process of multiplication, the resulting complex number, and some interesting properties.
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.
The Multiplication Process
To multiply complex numbers, we use the distributive property, just like with real numbers. Let's break down the multiplication step by step:
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First two numbers: (1+i)(2+3i) = 1(2+3i) + i(2+3i) = 2 + 3i + 2i + 3i² = -1 + 5i (since i² = -1)
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Next two numbers: (3+4i)(4+5i) = 3(4+5i) + 4i(4+5i) = 12 + 15i + 16i + 20i² = -8 + 31i
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Final multiplication: (-1 + 5i)(-8 + 31i) = -1(-8 + 31i) + 5i(-8 + 31i) = 8 - 31i - 40i + 155i² = -147 - 71i
The Resulting Complex Number
Therefore, the product of (1+i)(2+3i)(3+4i)(4+5i) is -147 - 71i. This is a complex number with a real part of -147 and an imaginary part of -71.
Geometric Interpretation
Complex numbers can be represented graphically on a complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. Each multiplication by a complex number can be viewed as a rotation and scaling in this plane. The resulting complex number -147 - 71i represents a point in this plane.
Conclusion
The multiplication of complex numbers (1+i)(2+3i)(3+4i)(4+5i) demonstrates the distributive property and results in the complex number -147 - 71i. This process highlights the interesting properties of complex numbers, including their ability to be represented geometrically and their use in various mathematical and scientific fields.