Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$(5+\sqrt{-27})-(-12+\sqrt{-48})$$

Step-by-Step Solution

Answer

The result of the addition/subtraction of the two given complex numbers is \(-7 + 7\sqrt{3}i\).

Step 1: Rewrite the problem using imaginary numbers

First, rewrite the exercise with imaginary numbers, as \((5 + 3\sqrt{3}i) - (12 - 4\sqrt{3}i)\). That's because \(\sqrt{-27}= \sqrt{27}\cdot i = 3\sqrt{3}\cdot i\) and \(\sqrt{-48}= \sqrt{48}\cdot i = 4\sqrt{3}\cdot i\).

Step 2: Perform the subtraction

Subtract the terms by performing subtraction of corresponding real parts and imaginary parts separately, giving us \((5 - 12) + (3\sqrt{3}i - -4\sqrt{3}i)\) = \(-7 + 7\sqrt{3}i\).

Step 3: Write the final result

The result, \(-7 + 7\sqrt{3}i\), is already in standard form for complex numbers, \(a+bi\).

Key Concepts

Imaginary Numbers

Imaginary numbers might sound complicated, but they're quite simple once you get to know them! They'll become your helpful sidekick when you're dealing with square roots of negative numbers. In math, we know that squares of any real numbers give us positive outcomes. So what do we do when we encounter calculators screeching errors at us with the square root of a negative number? That's where imaginary numbers come to the rescue. These numbers use the unique unit called 'i'.

• 'i' is defined as the square root of -1, making it a special number that helps solve otherwise impossible problems.
• For example, if you see \(\sqrt{-4}\), you can rewrite it as \(2i\) because it's \(\sqrt{4}\times i\).

By using imaginary numbers, you can break down complex square roots into more manageable parts so you can continue with real-world math operations! Remember, 'i' always appears in its simplest form and paves the way for entering the world of complex numbers.

Standard Form in Complex Numbers

The standard form of complex numbers is easy to grasp and provides a straightforward way to express these characters of mathematics. Complex numbers are a mix of real and imaginary numbers, and the standard form shows this perfectly. If you like things neat and orderly, then you'll love the standard form!

• The standard form for any complex number is written as \(a + bi\), where:
• 'a' is the real part of the complex number.
• 'bi' is the imaginary part where 'b' is a real number coefficient for 'i'.
• For example, in the complex number \(3 + 4i\), 3 is the real part, and \(4i\) is the imaginary part.

When dealing with complex numbers, writing them in this format allows you to handle operations such as addition and subtraction more easily. Plus, this form keeps things tidy when presenting your results, almost like having your math library organized in its own magical way.

Addition and Subtraction of Complex Numbers

Manipulating complex numbers is much like handling regular numbers, but with a neat twist: managing both real and imaginary parts. When adding or subtracting complex numbers, you need to treat the real and imaginary parts as separate components, conducting operations on each part independently.

• For addition: Simply add together the real parts separately from the imaginary parts.
• For subtraction: Subtract the real parts and then subtract the imaginary parts accordingly.

Consider adding \((3 + 2i) + (1 + 4i)\). You'll add the real parts \(3 + 1 = 4\) and the imaginary parts \(2i + 4i = 6i\), resulting in \(4 + 6i\).
Similarly, for subtraction such as \((5 + 3i) - (2 + i)\), subtract the real components \(5 - 2 = 3\) and the imaginary parts \(3i - i = 2i\) to get \(3 + 2i\).
Handling complex numbers like these lets you seamlessly perform arithmetic while respecting all components. This method ensures clarity and accurate results, offering a rewarding crunch of numbers under your fingertips.