Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$(7+\sqrt{-18})+(3+\sqrt{-32})$$

Step-by-Step Solution

Answer

The result in standard form is \(10 + 7\sqrt{2}i\).

Step 1: Simplify the Imaginary Numbers

Start by simplifying \(\sqrt{-18}\) and \(\sqrt{-32}\). The negative sign in the square root can be simplified to 'i' which is the imaginary unit. So, \(\sqrt{-18} = \sqrt{18} * i = 3\sqrt{2} * i\) and \(\sqrt{-32} = \sqrt{32} * i = 4\sqrt{2}*i\).

Step 2: Substitute the Simplified Imaginary Numbers

Substitute the simplified imaginary numbers back into the expression. So, \((7 +\sqrt{-18}) + (3 + \sqrt{-32}) = (7 + 3\sqrt{2}i) + (3 + 4\sqrt{2}i)\)

Step 3: Perform the Addition

Add the real parts together and then add the imaginary parts together. So, \((7 + 3\sqrt{2}i) + (3 + 4\sqrt{2}i) = (7+3) + (3\sqrt{2}i + 4\sqrt{2}i) = 10 + 7\sqrt{2}i\)

Key Concepts

Imaginary Unit

In mathematics, an imaginary unit is a special number used to simplify the handling of square roots of negative numbers. The imaginary unit is commonly denoted by the symbol 'i', where the primary relationship is defined by the property that the square of 'i' equals -1. Thus, we have:

• i = \(\sqrt{-1}\)
• i² = -1

This concept is critical when dealing with square roots of negative numbers, which arise in a variety of mathematical contexts. Instead of dealing directly with \(\sqrt{-x}\) (where x is a positive number), we rewrite it using the imaginary unit as \(\sqrt{x} \cdot i\). This simplifies the computation and ensures we maintain consistent mathematical rules.
In the original exercise, the imaginary units appear when simplifying expressions like \(\sqrt{-18}\) and \(\sqrt{-32}\). By utilizing 'i', these expressions were transformed into \(3\sqrt{2}i\) and \(4\sqrt{2}i\), making further calculations more manageable.

Addition of Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, written in the form \(a + bi\), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The original exercise required us to add complex numbers together. To perform this addition, you simply add the real parts together and the imaginary parts together separately.
Here's how you do it:

• Real Part: If you have two complex numbers, \((a + bi)\) and \((c + di)\), the real parts are 'a' and 'c'. Simply add them together: \(a + c\).
• Imaginary Part: For the imaginary parts, you add up 'b' and 'd', giving you: \((b + d)i\).

In the step-by-step solution given, we see this method in action. Combining the numbers \((7 + 3\sqrt{2}i)\) and \((3 + 4\sqrt{2}i)\), the individual real parts '7' and '3' are added to get '10', and the imaginary parts are added as \(3\sqrt{2}i + 4\sqrt{2}i = 7\sqrt{2}i\). Thus, the result is \(10 + 7\sqrt{2}i\).

Simplifying Square Roots

Square roots are often seen as daunting in equations, especially when dealing with negatives. However, simplifying square roots makes complex numbers much easier to handle. When simplifying square roots of negative numbers, the imaginary unit 'i' becomes essential.
To simplify square roots of negatives, follow these steps:

• Identify the square root of a negative number, for instance \(\sqrt{-x}\).
• Break it into the product of the square root of the positive part and the imaginary unit 'i': \(\sqrt{x} \cdot i\).
• Simplify \(\sqrt{x}\) if possible.

For instance, in the exercise, \(\sqrt{-18}\) is simplified by recognizing it as \(\sqrt{18} \cdot i\). Then, \(\sqrt{18}\) is further broken down to \(3\sqrt{2}\), leading to the complete simplification: \(3\sqrt{2}i\).
The same process is applied to \(\sqrt{-32}\), resulting in \(4\sqrt{2}i\). By consistently applying these simplification techniques, complex number computations become much straightforward.