Question

Multiply and simplify. $$7 \times 2 i$$

Short answer

The product is 14i.

Step-by-step solution

Understanding Complex Numbers Multiplication

When multiplying a real number by an imaginary number, simply multiply the real number by the imaginary part.

Perform the Multiplication

Multiply the real number 7 by the imaginary unit 2i: \( 7 \times 2i = 14i \). The result is an imaginary number.

Final Simplification

There is no further simplification required since 14i is already in its simplest form.

Key concepts

Imaginary Numbers

Imaginary numbers are an extension of the real number system where solutions to certain equations that cannot be solved using only real numbers are found. For example, the square root of a negative number isn't defined in the system of real numbers, as no real number squared gives a negative result.

The unit imaginary number is denoted as 'i', and by definition, it is the square root of -1, or in mathematical terms: \(i = \sqrt{-1}\). Any imaginary number can be written as \(bi\), where 'b' is a real number. When dealing with multiplication involving imaginary numbers, remember that multiplying the imaginary unit 'i' with itself results in \(i^2 = -1\).

This property is critical when simplifying expressions with imaginary numbers, as it frequently requires converting \(i^2\) back to -1 to keep the numbers in a workable format.

Real Numbers

In contrast to imaginary numbers, real numbers consist of all the numbers on the number line, including all the rational and irrational numbers. Rational numbers include integers, whole numbers, and fractions that can be written as a ratio of two integers. Irrational numbers cannot be written as simple fractions - examples include \(\pi\) and \(\sqrt{2}\).

When we multiply a real number by an imaginary number, we stay within the scope of real number arithmetic, simply using the imaginary unit as a placeholder to indicate that the product is on a different 'axis' of the complex plane. This multiplication does not fundamentally alter how we handle the real number; it just tags the result with an 'i' to show it is an imaginary product.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that minimizes the complexity of an expression without changing its value. The aim is to make the expression as straightforward as possible. This could involve combining like terms, factoring, canceling common factors within fractions, and reducing expressions to their lowest terms.

When dealing with complex numbers, simplifying often means ensuring that any powers of the imaginary unit 'i' are reduced to 'i', '-i', '1', or '-1'. For the multiplication of a real number and an imaginary number, the simplification process is often straightforward since all we do is multiply the real number coefficient with the imaginary unit, just as we did in the example given in the original exercise, \( 7 \times 2i = 14i \).