Introduction to Engineering Mathematics

Summary:

This text provides an introduction to engineering mathematics and calculus, focusing on numbers and functions. It starts by exploring the different types of numbers, ranging from positive and negative integers to fractions and rational numbers. The concept of irrational numbers is introduced through the square root of two, highlighting its uniqueness and the difficulty in precisely defining numbers. The real number line is introduced as a visual representation of real numbers, and intervals are explained as subsets of real numbers.

The text also delves into the concept of functions, defining them as rules that assign a value to each input. Linear functions are introduced, along with their graphs and properties, such as slope and y-intercept. Implicit functions, defined by equations rather than explicit formulas, are also discussed. The idea of inverse functions is introduced, emphasizing their relationship with the original functions and their potential to undo their effects.

Excerpt:

Introduction to Engineering Mathematics

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1.2. A reason to believe in √
2. The Pythagorean theorem says that the hypotenuse of a right triangle with sides 1 and 1 must be a line segment of length 2. In middle or high school, you learned something similar to the geometric construction of a line segment whose length is 2. Take a square with a side of length 1, and construct a new square, one of whose sides is the diagonal of the first square. The figure you get consists of 5 triangles of equal area, and by counting triangles, you see that the larger square has exactly twice the area of the smaller square. Therefore the diagonal of the smaller square, being the side of the larger square, is 2 as long as the side of the smaller square.

Why are real numbers called real? All the numbers we will use in this first semester of calculus are “real numbers.” At some point (in 2nd-semester calculus), it becomes useful to assume that there is a number whose square is −1. No real number has this property since the square of any real number is positive, so it was decided to call this new imagined number “imaginary” and to refer to the numbers we already have (rationals, 2-like things) as “real.”