Complex Number To Standard Form (-10 + √-40) / 2 Step By Step

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This comprehensive guide will walk you through the process of expressing the complex number (-10 + √-40) / 2 in standard form, which is a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Understanding standard form is crucial for performing various operations with complex numbers, such as addition, subtraction, multiplication, and division. We will break down each step with clear explanations and examples, ensuring you grasp the underlying concepts thoroughly. Let’s dive in and master the art of writing complex numbers in their standard form.

Understanding Complex Numbers and Standard Form

Before we tackle the problem, it’s essential to understand what complex numbers are and why standard form is important. A complex number is a number 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., i = √-1). The a part is called the real part, and the bi part is called the imaginary part. The standard form, a + bi, allows us to easily compare and perform operations on complex numbers.

The importance of standard form in complex number manipulation cannot be overstated. When a complex number is in standard form, it becomes straightforward to identify its real and imaginary components. This identification is critical for performing arithmetic operations like addition, subtraction, multiplication, and division. For example, when adding or subtracting complex numbers, you simply add or subtract the real parts and the imaginary parts separately. Similarly, multiplication and division operations benefit from the clear structure provided by the standard form, allowing for the application of the distributive property and conjugate methods more effectively. Furthermore, expressing complex numbers in standard form simplifies graphical representation on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part on the vertical axis. This visual representation enhances understanding and makes it easier to solve problems related to complex numbers in various fields such as electrical engineering, quantum mechanics, and fluid dynamics. Therefore, mastering the conversion to standard form is a foundational skill in complex number arithmetic and analysis.

Step-by-Step Solution

Now, let’s convert the given complex number, (-10 + √-40) / 2, into standard form. We'll break this down into manageable steps:

Step 1: Simplify the Square Root of a Negative Number

The first step involves simplifying the square root of the negative number, √-40. Recall that √-1 is defined as i. We can rewrite √-40 as follows:

√-40 = √(40 * -1) = √40 * √-1

Now, we simplify √40. We look for perfect square factors of 40. The largest perfect square factor of 40 is 4:

√40 = √(4 * 10) = √4 * √10 = 2√10

So, combining these steps, we have:

√-40 = 2√10 * i = 2i√10

Step 2: Substitute the Simplified Radical Back into the Expression

Now that we've simplified √-40 to 2i√10, we substitute this back into the original expression:

(-10 + √-40) / 2 = (-10 + 2i√10) / 2

This substitution is crucial because it replaces the square root of a negative number with its equivalent expression using the imaginary unit i. By doing so, we transform the expression into a form that is more amenable to simplification and conversion into standard form. The importance of this step lies in its ability to bridge the gap between a seemingly complex term involving a square root of a negative number and a more manageable form that can be easily manipulated. It sets the stage for the subsequent steps, where we will separate the real and imaginary parts, ultimately leading to the expression of the complex number in standard form. This transformation not only simplifies the immediate expression but also makes it easier to perform operations with complex numbers in later stages, highlighting its significance in the overall process.

Step 3: Divide Both Terms in the Numerator by the Denominator

Next, we divide both the real and imaginary terms in the numerator by the denominator, which is 2:

(-10 + 2i√10) / 2 = -10/2 + (2i√10)/2

This step is essential as it separates the complex number into its real and imaginary components, which is a key characteristic of the standard form a + bi. By dividing each term in the numerator by the denominator, we isolate the real part and the imaginary part, making it easier to identify and simplify them individually. This process not only clarifies the structure of the complex number but also prepares it for the final arrangement in standard form. The separation allows for a clearer understanding of the complex number's position on the complex plane, where the real part corresponds to the x-coordinate and the imaginary part corresponds to the y-coordinate. This division is a fundamental algebraic manipulation that ensures the complex number is expressed in a way that facilitates further analysis and operations, underscoring its importance in converting to standard form.

Step 4: Simplify the Fractions

Now, we simplify each fraction:

-10/2 = -5

(2i√10) / 2 = i√10

The simplification of these fractions is a critical step in expressing the complex number in its most concise and understandable form. By performing basic arithmetic, we reduce the real and imaginary parts to their simplest equivalents, making the standard form of the complex number clearer. The simplification not only tidies up the expression but also facilitates easier comparison with other complex numbers and enables more straightforward calculations in subsequent operations. For example, reducing -10/2 to -5 gives an immediate, clear value for the real part of the complex number, and similarly, simplifying (2i√10) / 2 to i√10 clarifies the imaginary component. This process of simplification enhances the usability of the complex number in various mathematical contexts, highlighting its significance in the overall conversion to standard form.

Step 5: Write the Complex Number in Standard Form

Finally, we write the simplified expression in the standard form a + bi:

-5 + i√10

Here, a = -5 and b = √10. This is the standard form of the given complex number.

Common Mistakes to Avoid

When converting complex numbers to standard form, there are a few common mistakes to watch out for:

  1. Incorrectly Simplifying Square Roots of Negative Numbers: Always remember to factor out √-1 as i before simplifying further.
  2. Forgetting to Distribute the Denominator: When dividing a complex expression by a number, make sure to divide both the real and imaginary parts.
  3. Not Simplifying Fractions: Ensure that all fractions are simplified to their lowest terms.

Avoiding these common pitfalls will ensure that you accurately convert complex numbers into standard form.

Practice Problems

To solidify your understanding, try converting the following complex numbers into standard form:

  1. (6 + √-16) / 2
  2. (4 - √-25) / 5
  3. (-8 + √-12) / 4

Working through these practice problems will reinforce your ability to apply the steps we’ve discussed and help you become more confident in handling complex numbers.

Conclusion

In conclusion, expressing the complex number (-10 + √-40) / 2 in standard form involves simplifying the square root of the negative number, substituting the simplified radical back into the expression, dividing both terms in the numerator by the denominator, simplifying the fractions, and writing the result in the form a + bi. By following these steps carefully, we have successfully converted the given complex number into its standard form: -5 + i√10. Mastering this process is essential for working with complex numbers in various mathematical and scientific contexts. Remember to avoid common mistakes and practice regularly to build your skills. With a solid understanding of standard form, you'll be well-equipped to tackle more complex problems involving complex numbers.

By practicing these steps and understanding the underlying principles, you can confidently convert any complex number into its standard form. This skill is essential for further studies in mathematics, physics, engineering, and other related fields.