91Ó°ÊÓ

Real numbers, denoted by the symbol \(\backslashmathbb{R}\), include all the numbers that can be located on the number line. These encompass:

• Natural numbers (1, 2, 3,...)
• Whole numbers (0, 1, 2,...)
• Integers (...-2, -1, 0, 1, 2,...)
• Rational numbers (like fractions, \(\frac{1}{2}\text{ or }0.75\))
• Irrational numbers (numbers that can't be expressed as fractions, like \(\backslashsqrt{2}\text{ or }\backslashpi\))

One critical aspect of real numbers is that they do not include any imaginary or complex parts. For example, when we say \(4-5i\text{ is not in }\backslashmathbb{R}\), it is because \(4-5i\) contains an imaginary part (-5i), which disqualifies it from being classified as a real number.

Imaginary numbers are numbers that involve the imaginary unit \(\backslashmathbf{i}\), which is defined as \(\backslashmathbf{i}^2 = -1\). These are essential in extending the number system so that not all equations remain unsolvable. For instance:

• The square root of -1 is written as \( \backslashsqrt{-1} = \backslashmathbf{i} \)
• Numbers like \(2i, \text{ and }-5i\) are purely imaginary numbers.

Imaginary numbers are integral in the field of complex numbers and are used in various advanced fields like engineering and physics to solve problems involving waveforms, oscillations, and much more.

A complex number is any number that can be written in the form \(a + bi\), where \(\backslashmathbf{a}\) and \(\backslashmathbf{b}\) are real numbers, and \(\backslashmathbf{i}\text{ represents the imaginary unit, satisfying } \backslashmathbf{i}^2 = -1\). This means all complex numbers have a real part (\backslashmathbf{a}) and an imaginary part (\backslashmathbf{bi}).
For example, \(4-5i\text{ is a complex number where} a = 4 \text{ and } b = -5\).
Important properties of complex numbers include:

• They encompass real numbers when the imaginary part is zero (e.g., \(7 = 7 + 0i\)).
• They also include purely imaginary numbers (e.g., \(-3i\)).

As shown in the exercise, \(4-5i\) is a complex number since it fits \(a + bi\text{ with }a=4 \text{ and }b=-5\). Complex numbers can be added, subtracted, and multiplied using algebraic rules, which gives a versatile tool in many areas of science and engineering.