91Ó°ÊÓ

Understanding complex numbers is crucial in the study of mathematics, especially when dealing with equations that have no real number solutions. Complex numbers extend the idea of the one-dimensional number line into a two-dimensional complex plane, by including the imaginary unit 'i', which is defined by its property that \(i^2 = -1\).

A complex number is written in the standard form \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part. Here, \(b\) is a real number, and \(i\) is the imaginary unit. For example, \(3 + 4i\) or \(2 - 7i\) are both complex numbers. In our exercise, the complex number in standard form is written as \(0 + 0.02i\), emphasizing that even if the real part is zero, the number is still complex.

The imaginary unit, denoted as \(i\), is a mathematical concept that allows the extension of real numbers to include solutions to square roots of negative numbers. This concept is vital because it fills a gap in the real number system, which does not have real solutions for square roots of negative numbers.

By definition, \(i\) is the square root of -1, that is, \(i^2 = -1\). It enables us to perform calculations that would otherwise be impossible with just real numbers. For example, without the imaginary unit, we cannot find a real number solution for \(\sqrt{-4}\), but with the imaginary unit, we can express this as \(2i\). In our example, we used the imaginary unit to express \(\sqrt{-0.0004}\) as \(0.02i\).

Square roots are operations that ask, 'What number squared gives us a particular value?' The square root of \(x\), denoted as \(\sqrt{x}\), is a value that, when multiplied by itself, yields \(x\). When \(x\) is a positive real number, its square root is also a real number. However, when \(x\) is negative, its square root is an imaginary number.

In our exercise, we encountered the square root of a negative number, \(\sqrt{-0.0004}\). To solve this, we first recognize that the square root operation can be separated into the product of the positive square root and \(i\), dealing with the negative part. The positive square root of 0.0004 is 0.02, and for the negative part, we have \(i\), hence \(\sqrt{-0.0004} = 0.02i\). This operation demonstrates the crucial link between square roots and the imaginary unit in forming complex numbers.