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Chapter 2: Problem 6

In Exercises 1 to 10 , write the complex number in standard form. $$\sqrt{25}+\sqrt{-9}$$

Short Answer

Expert verified
The complex number in standard form from the given expression is \(5 + 3i\).

Step by step solution

01

Calculate the square roots

Calculate the square root of the two numbers. The square root of 25 is 5. The square root of -9 is \(\sqrt{(-1) \times 9}\) which gives \(3i\) as the square root of \(-1\) is \(i\).
02

Write in standard form

Write the result in standard form. The result of the operation is \(5 + 3i\). The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Therefore, the number in standard form would be \(5 + 3i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form
Complex numbers can often seem confusing, but understanding the standard form simplifies things a lot. The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. In this expression, \(a\) is known as the real part, and \(b\) is the imaginary part.
For example, let's consider the complex number formed by \(5 + 3i\). Here, 5 is the real part and 3 is the imaginary part.
  • Always ensure to write them in the form of \(a + bi\) where both \(a\) and \(b\) are evident.
  • This format helps clearly separate contributions from real and imaginary numbers, making calculations easier.
Practicing converting numbers into this form can improve understanding and ease operations with complex numbers.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in dealing with complex numbers. The key property of \(i\) is such that \(i^2 = -1\). This means \(i\) represents the square root of negative one. When faced with finding the square root of a negative number, the imaginary unit becomes particularly useful.
The expression \(\sqrt{-9}\) can be rewritten as \(\sqrt{(-1) \times 9}\), which simplifies to \(3i\). Here, the \(i\) is crucial as it allows us to work with square roots of negative values.
  • Remember, any real number multiplied by \(i\) creates the imaginary part.
  • Using \(i\) correctly transforms impossible calculations into solvable ones.
Understanding \(i\) helps bridge the gap between real and complex number calculations.
Square Roots
Square roots are essential in simplifying expressions like the one in our exercise, \(\sqrt{25} + \sqrt{-9}\). While the square root of a positive number, such as \(\sqrt{25}\), results in a straightforward real number (in this case, 5), the square root of a negative number involves the imaginary unit \(i\).
When calculating \(\sqrt{-9}\), the negative inside signals the need for \(i\). You rewrite it as \(\sqrt{(-1) \times 9}\), resulting in \(3i\). Thus, the number conforms to the complex number format.
  • For positive numbers, follow conventional square root rules.
  • For negative numbers, integrate the imaginary unit.
By mastering the square root process for both positive and negative roots, working with complex numbers becomes more intuitive.

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