Problem 36 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ -2 i \cdot-11 i $$

Step-by-Step Solution

Answer

The product is \(-22\).

Step 1: Apply the Multiplication Rule for Complex Numbers

When multiplying two imaginary numbers in the form of \(a \, i\), we use the formula \(a \, i \, \cdot \, b \, i = ab \, i^2\). Here, the numbers are both pure imaginary, so we multiply their coefficients.

Step 2: Multiply the Coefficients

Multiply the coefficients -2 and -11 to get: \((-2) \times (-11) = 22\).

Step 3: Apply the Imaginary Unit Rule

Remember that \(i^2 = -1\). Since we initially have \(i^2\) from \(-2i \cdot -11i\), we multiply 22 by -1: \((22 \cdot i^2 = 22 \cdot (-1) = -22)\).

Step 4: Write in the Form \(a + bi\)

Combine the result to express in the form \(a + bi\): The result is purely real as \(-22 + 0i\). Thus, the answer is \(-22\), which simplifies to just \(-22\) since there is no imaginary component.

Key Concepts

Imaginary Numbers

Imaginary numbers can seem like a mysterious concept at first, but they are actually quite simple. They are numbers that have a real component of zero and a complex component involving the square root of -1, often represented as the letter "i". This gives us the basic building block for imaginary numbers: if you see an 'i' in a number's expression, that means you're dealing with its imaginary part.
Let's break it down: if you have a number like 3i, it's purely imaginary because it doesn’t have any real part. This aspect of imaginary numbers is crucial for solving more complex problems involving complex numbers.
Understanding imaginary numbers opens a new dimension in mathematics, allowing us to solve equations that don't have real solutions. They are indispensable in fields like engineering and physics where calculations often involve electrical circuits or wave patterns. Imaginary numbers help handle these complex ideas easily.

Multiplying Complex Numbers

Multiplying complex numbers can be approached by applying the distributive property, similar to multiplying binomials. When two complex numbers involve imaginary parts, the calculation gets even more interesting.
For example, when multiplying \(-2i imes -11i\), you should first multiply the coefficients: \(-2\) and \(-11\) multiply to give \(22\). Then you multiply the imaginary parts: \(i imes i = i^2\).
This leads to a product \(22i^2\), which needs further simplification. What’s crucial here is knowing what \(i^2\) represents, leading us naturally to our final section.
Remember, multiplying complex numbers may give you a result that is either complex or purely real depending on the components involved. The order of operations, dealing separately with real and imaginary parts, is essential to solve these problems accurately.

Imaginary Unit

The imaginary unit, often denoted by the letter 'i', is a foundational concept in complex numbers. It is defined by the property that \(i^2 = -1\). This property allows imaginary numbers to extend the number system beyond the real numbers.
Here's why \(i^2 = -1\) matters: if you multiply an imaginary number by itself, it transforms into a real number, specifically into the negative form. This means when faced with expressions like \(22i^2\), as we saw earlier, knowing \(i^2\) equals \(-1\) helps convert it quickly into a real number, \(-22\).
Thus, the imaginary unit is not only a concept but an essential tool to navigate through complex number operations, lending depth and breadth to mathematical problem-solving.