which subtraction expression has the difference 1 + 4i

Guri Bee

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23-10-2024

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Unmasking the Complex Subtraction: Finding the Missing Equation

In the realm of complex numbers, subtraction takes on a unique character. Instead of simply subtracting magnitudes, we work with both real and imaginary components. This article explores the challenge of finding a subtraction expression that yields the difference 1 + 4i.

Understanding Complex Numbers

Complex numbers, often denoted by the letter 'z', consist of two parts: a real part and an imaginary part. They are expressed in the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part, and 'i' is the imaginary unit, defined as √-1.

The Quest for the Subtraction Equation

Our goal is to discover a subtraction equation that results in the complex number 1 + 4i. This means we need to find two complex numbers, let's call them z1 and z2, such that:

z1 - z2 = 1 + 4i

Approaching the Solution

Here's where the magic of complex numbers comes into play. To subtract complex numbers, we simply subtract their real parts and their imaginary parts separately.

Let's assume:

• z1 = a + bi
• z2 = c + di

Then, our subtraction equation becomes:

(a + bi) - (c + di) = (a - c) + (b - d)i = 1 + 4i

This gives us two equations:

1. a - c = 1
2. b - d = 4

Finding Solutions

Now, we have a system of equations. There are infinitely many solutions for 'a', 'b', 'c', and 'd' that satisfy these equations. Here's one example:

• Let a = 2 and c = 1. This satisfies the first equation (2 - 1 = 1).
• Let b = 6 and d = 2. This satisfies the second equation (6 - 2 = 4).

Therefore, one possible subtraction equation is:

(2 + 6i) - (1 + 2i) = 1 + 4i

Generalizing the Solution

The solution above is just one example. We can find countless other subtraction expressions that satisfy the condition. Any pair of complex numbers that satisfy the equations a - c = 1 and b - d = 4 will work.

Conclusion

Unmasking the subtraction expression leading to 1 + 4i involves understanding how complex numbers are subtracted. By setting up a system of equations based on the real and imaginary components, we can find multiple solutions, showcasing the versatility and richness of the complex number system.

Attribution:

This article draws inspiration from the helpful discussions on GitHub, where numerous contributors share their insights on complex numbers. Thank you to the entire GitHub community for their valuable contributions.