## Understanding Calculus

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 Preface 1. Why Study   Calculus 2. Numbers 3. Functions 4. The Derivative 5. Differentiation 6. Applications 7. Free Falling   Motion 8. Understanding  Derivative 9. Derivative  Approximations 10. Integration   Theory 11. Understanding   Integration 12. Differentials

 Inverse Functions Exponents Exponential  Functions Applications of   Exponential  Functions Sine and Cosine   Function Sine Function Sine Function -   Differentiation and   Integration Oscillatory Motion Mean Value   Theorem Taylor Series More Taylor Series Integration   Techniques

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Numbers and their uses

### Chapter 2 - Understanding Numbers

#### Section 2.1 - Numbers and their Uses

What is a number? What do all those abstract symbols floating about really mean? To begin with, we can define the number as an existence. So if I had a few tennis balls, the number of balls would simply refer to how many individual tennis balls I am actually holding. 4 balls would mean four existences or :

2 balls would mean

A number therefore does not depend on any specific quality of the object. 10 balls and 10 eggs are the same number of things.

Now that we have defined the number as something created by the human mind to count objects, then let us proceed to examine the Laws of Arithmetic.

The laws of arithmetic are based on the addition and multiplication of numbers. But what does it mean to add two or three numbers together? To begin with addition assumes that all your numbers refer to something and that thing is the same throughout. So 3 pens added with 3 pens is six pens. We are bringing together all our things and then counting the sum. So:

In multiplication we are doing something a bit different, we are creating multiples of a number. When we multiply two numbers x and y : we are creating y multiples of x which is the same as adding x with itself y times. This can also be thought of as a grid of x rows and y columns where the product of the numbers is the sum of all the objects inside the grid. So 4 * 6 is:

In this example we only added 4 with itself 6 times and summed up the number we got. This is all that multiplication is about, creating multiples of a number or just repetitive addition. Multiplication of two numbers x and y can therefore be expressed as:

or 3*7 is: 7 + 7 + 7

You are probably asking what is the use of such an operation or why is multiplying numbers an important thing in mathematics. It is fairly obvious how it is used when a baker wants to find the total cost of 42 cupcakes at \$2/cupcake. The answer is 42*2. But is not so easy to understand for example why in Newton's second Law of Motion:

or why interest earned in money is:

The reason multiplication dominates these equations and so many others depends entirely on the context of the equation and where it is derived from. As we study many equations in this book the difference between multiplication and addition will become clearer.

We have studied integers which refer to full existences such as 2 books, 4 cats and 110 bananas. But what about half a candy bar or a third of a day? When the existence or object is broken up into equal parts we say we are dividing and we use fractions to represent the division. Division is essentially the opposite or inverse operation of multiplication. Whereas multiplication creates equal sets of a number, division creates equal sub-sets within the number. The fractions that are used to represent division are often written as: where the bottom number refers to how many pieces the object has been broken into. For example the following rod has been broken into 8 partitions where each small block is: ’th of the original object:

We can now define x divided by y to be where y indicates the number of equal parts an object is subdivided into and x indicates how many parts you have. The relationship between multiplication and division is best expressed by the equation, If there is an x on top of the y it tells us that there are x times or x parts of

Whereas division creates sub-sets within an object, multiplication creates sets outside the object. Dividing an object by another number x, creates x sub-sets within the object such that multiplying one sub-set by x give the original object again. For this reason multiplication and division are called inverse operation of each other.

We can now state the five laws of arithmetic:

1) A + B = B + A

2) A + ( B + C)= ( A + B) + C

3) AB = BA

4) A(BC) = (AB)C

The fourth property is saying that multiplication of more than two numbers is independent of the order they are multiplied in.

5) A( B + C)= AB + AC

To this point our discussion of numbers has been limited to positive numbers but what about negative numbers? Before discussing what it means to have -3 books let us look at the number 0. Just as numbers define an amount of something, so we must have a way of defining nothing. Zero, therefore represents nothing or no-existence. It is just as much a number as 6 or 7 is. For example:

As we shall study, zero is an important number as it helps in solving equations and defining infinity.

Negative numbers, in the simplest sense, are used to define those numbers that are the result of a larger number being subtracted from a smaller one. Imagine a man who has six pens. A young man comes to buy eight pens. How many are left? The answer is not 0 but -2. The shopkeeper sells all six, but he still needs two to give to the man. Negative numbers can therefore be thought of as numbers that do not exist but should exist. They become positive when they take on a physical existence otherwise they are missing pieces in a puzzle.

Our discussion of numbers has been limited to objects. This need not be the case. Numbers can be used to represent abstract existences such as 2 days, 3 years, 4 classes, etc. Recall how we defined the number to be meaningless without referring to something. Three books meant one book + one book + one book. The number can also be thought of as a repetition of an occurrence. A day is defined as the rising of the sun, setting, then rising again. This “ action” is cyclical and continues indefinitely . We can therefore refer to each cycle of the sun rising as one day and four days would then represent four cycles of the sun rising.

Here we have extended the definition of the number to not only include objects but also actions. The difference between an action and an object is a difficult one to explain. While an object represents something we can see, touch, hold and feel, an action is only that which happens. Actions refer to change or movement. Therefore, an action's existence is defined by what takes place during that time frame of a beginning and an end.

Numbers are not limited to objects and actions alone, they can also be used to count ideas and other abstract concepts that exist only in the mind. Simply by existing in the mind means it can be counted. When somebody says I have two things to say to you, each thing is made up of a different choice of words, phrases, not to mention the action or object it may refer to.

To conclude our discussion of the number here is a short definition.

Number - A symbol used to express an object's, action's or abstraction's repetitive nature.

Next section -> Section 2.2 - Interpreting Numbers

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