Question

\((7 i)(-3 i)\)

Short answer

The result of \((7i)(-3i)\) is 21.

Step-by-step solution

Identification

Identify the given complex numbers, which are \(7i\) and \(-3i\). Notice that both are imaginary numbers.

Multiplying the complex numbers

Once you multiply them, \(7i * -3i = -21i^2\).

Simplifying product

Remember that \(i^2\) is \(-1\), hence the multiplication becomes \(-21 * -1 = 21\).

Key concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. This can be represented in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The letter \( i \) represents the imaginary unit.
When working with complex numbers, it's important to understand that they expand our number system beyond the real numbers. Some key points about complex numbers include:

• Complex numbers allow us to solve equations that have no real solutions, such as \( x^2 + 1 = 0 \).
• They are used in various fields including engineering, physics, and computer science for handling periodic signals and waves.
• Every real number can be considered a complex number with an imaginary part of zero.

Using complex numbers can make calculations involving roots and variables much simpler, yet they require understanding of the imaginary unit \( i \) and its properties.

Imaginary Numbers

Imaginary numbers are centered around the concept of providing a solution for equations that don't have real solutions. The simplest imaginary number is \( i \), which is defined as the square root of \( -1 \) (\( i = \sqrt{-1} \)). This means multiplying \( i \) by itself (\( i \times i \)) gives \( -1 \).
Imaginary numbers can be confusing at first because visually imagining the square root of a negative number is challenging. However, they are practical and used extensively in complex number calculations:

• Multiplied with other imaginary numbers or real numbers, they form part of complex numbers.
• They are used to calculate oscillating forces and alternating current behavior in electronics.
• Their unique properties simplify many equations that would otherwise be hard to manage.

Once you get comfortable with the idea of \( i \) as an algebraic tool, manipulating imaginary numbers becomes more intuitive.

i-squared Identity

The \( i^2 \) identity is a crucial concept when working with complex numbers. It simplifies calculations involving imaginary numbers by using the identity \( i^2 = -1 \). This identity helps to transform expressions that contain \( i^2 \) into real numbers, facilitating simplification and problem-solving.
For instance, in the multiplication of complex numbers like \(-21i^2\), we use the \( i^2 = -1 \) identity to get:

• Convert \(-21 i^2 \) to \(-21 \times (-1)\), leading to \( 21 \).
• This effectively transforms an otherwise complex expression into a manageable real number.
• Understanding this identity is pivotal for multiplying and simplifying expressions involving complex numbers.

This identity serves as a foundation for numerous complex number operations, solidifying how imaginary numbers integrate with real numbers for comprehensive problem-solving.