Simplifying Complex Expressions: A Guide to Understanding and Solving 1-i, 1-2i, ..., 1-10i

Introduction

In this article, we will guide you through simplifying a series of complex numbers. Specifically, we will walk through the steps to simplify the expression 1-i, 1-2i, 1-3i, ..., 1-10i. We will use fundamental concepts of complex numbers, including their magnitudes and the properties of the product of complex terms. This process will not only help you solve this particular problem but also provide a framework for understanding similar problems in the future.

Understanding the Problem

The given expression involves subtracting the imaginary unit i from a series of real numbers. The expression is:

1-i(1-2i)(1-3i)(1-4i)(1-5i)(1-6i)(1-7i)(1-8i)(1-9i)(1-10i)

Step 1: Rewriting Each Term

Each term of the form 1-ki can be rewritten using its polar form. We use the following identity:

1-ki sqrt{1 k^2}left(costheta_k isintheta_kright)

where k is an integer from 1 to 10, and theta_k is the angle such that tantheta_k k.

Step 2: Calculating the Magnitudes

The magnitude of each term is given by:

left|1-kiright| sqrt{1 k^2}

Hence, the product of the magnitudes is:

prod_{k1}^{10} left|1-kiright| sqrt{(1 1^2)(1 2^2)(1 3^2)(1 4^2)(1 5^2)(1 6^2)(1 7^2)(1 8^2)(1 9^2)(1 10^2)}

Step 3: Calculating the Product of the Terms

Now, we calculate the product of the terms:

1 1^2 2 1 2^2 5 1 3^2 10 1 4^2 17 1 5^2 26 1 6^2 37 1 7^2 50 1 8^2 65 1 9^2 82 1 10^2 101

The product of these values is:

2 times 5 times 10 times 17 times 26 times 37 times 50 times 65 times 82 times 101

Calculating this product step-by-step or using a calculator gives:

approx 2 times 5 10, 10 times 10 100, 100 times 17 1700, 1700 times 26 44200, 44200 times 37 1635400, 1635400 times 50 81770000, 81770000 times 65 5310050000, 5310050000 times 82 436620100000, 436620100000 times 101 44098630100000

Step 4: Final Product of Magnitudes

The final product of magnitudes is:

sqrt{44098630100000} approx 663000

Conclusion

Thus, the simplified expression is:

prod_{k1}^{10} (1-ki) sqrt{44098630100000} cdot e^{itheta}

where theta is the sum of the angles corresponding to each term. The exact angle can be computed but will be complex. While the exact complex form can be tedious to express without further computation, the product of magnitudes gives a good estimate of the size of the overall expression.

Note: For a practical solution, you can use a calculator like Google, WolframAlpha, or any scientific calculator to find the product more efficiently.