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Chapter 2: Problem 11

Write the complex number in standard form.$$8+\sqrt{-25}$$

Short Answer

Expert verified
The complex number in standard form is \(8 + 5i\).

Step by step solution

01

Identify the real and imaginary parts

In the complex number \(8+\sqrt{-25}\), the '8' is the real part and '\(\sqrt{-25}\)' is the imaginary part.
02

Convert the imaginary part into 'i' form

We know that \(\sqrt{-1}\) can be written as 'i'. So we can express \(\sqrt{-25}\) as \(5i\), since \(\sqrt{25}\) is 5.
03

Write the complex number in the standard form

Now, replacing \(\sqrt{-25}\) with \(5i\), then the whole expression becomes \(8 + 5i\), which is the standard form of a complex number.

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Key Concepts

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

Real Part of Complex Number
Complex numbers are made up of two parts: a real part and an imaginary part. The real part of a complex number is a regular number that you are already familiar with, like 8 in the complex number \(8 + \sqrt{-25}\).
Just like every familiar number, it lies on the real number line.
When written in standard form, the real part comes first.
  • It is independent of the imaginary unit \(i\).
  • The real part is a constant term.
  • It is denoted without any \(i\) next to it, unlike the imaginary part.
Understanding the real part separately is crucial because it helps in distinguishing it from the imaginary components. In calculations and graphing, the real part typically determines where the complex number lies on the x-axis of the complex plane.
Imaginary Unit
The imaginary unit \(i\) is a mathematical notion used to help work with numbers that result in a square root of negative numbers.
This might sound confusing because real numbers don't include negative square roots.
But in the context of complex numbers, it鈥檚 very useful.
  • The definition of the imaginary unit is \(i = \sqrt{-1}\).
  • When you have a negative square root like \(\sqrt{-25}\), it can be rewritten using \(i\).
  • This becomes \(5i\) because \(\sqrt{25} = 5\) and \(\sqrt{-1} = i\).
The imaginary unit allows us to represent numbers in a broader way. With \(i\), we can express and compute with numbers that would otherwise seem impossible, extending the scope of traditional arithmetic.
Standard Form of a Complex Number
The standard form of a complex number is very straightforward and neatly ordered.
This form is written as \(a + bi\), where \(a\) represents the real part and \(bi\) is the imaginary part.
As in the example \(8 + 5i\), standard form is used so it's easy to see both components at a glance.
  • It helps in easily performing operations such as addition, subtraction, and even more complicated ones like multiplication or division.
  • Writing complex numbers in this form simplifies comparison, graphing, and computation.
  • Each complex number has a unique representation in standard form.
Getting familiar with the standard form allows you to efficiently work with complex numbers, making them no less straightforward than any other types of numbers.

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Most popular questions from this chapter

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

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