Algebra/Arithmetic

From Wikibooks, open books for an open world
Jump to: navigation, search
Algebra
Algebra
 ← Who should read this book Arithmetic Order of Operations → 

Arithmetic is the process of performing certain operations on constant numbers or variables. There are seven arithmetic operations covered: addition, subtraction, multiplication, division, exponentiation, roots, and logarithm.

Addition[edit]

\ 1 + 1 = 2

To define the number one is a rather difficult task, but we all have a good intuitive sense of what "oneness" is. Oneness is the property of having or thinking of a singular quantity. For example, think of when you have one dollar, one bushel of potatoes, or one light year. From here we can recursively (that is, relating to the last one) define the natural numbers by assigning a new name for each new number of ones that we have:

1 1 one
2 1 + 1 two
3 1 + 1 + 1 three
n 1 + 1 + … + 1 n ones

Now that we have named the numbers we can define addition as the process of counting how many ones we have. For example,

\ 5 + 3 = (1 + 1 + 1 + 1 + 1) + (1 + 1 +1)  = 8

Addition Problems

Subtraction[edit]

Subtraction can likewise be defined as counting initial quantity of ones and removing some amount of them. For example:

\ 5 - 3 = (1+1+1+1+1) - (1+1+1) = 2 means 5 ones remove 3 ones, leaving a result of 2 ones.

Subtraction Problems

Multiplication[edit]

Multiplication is a shorthand (that is, a faster way of writing something) for repeated addition. For example:

\ 5\cdot3 = 15

What this means is to add up '3' 5 times; or add up '5' 3 times.

\ 5\cdot3 = 3 + 3 + 3 + 3 + 3 = 15

Multiplication Problems

Note that in some regions and cases, it is better to use the cross symbol ( \times ) or the letter "x" instead of the dot. Most regions use the dot instead of the cross symbol because the cross symbol looks like the letter "x".

Division[edit]

Division is the opposite of multiplication.

\ \frac{6}{3} = 2

This example asks if six is 1+1+1+1+1+1, and three is 1+1+1, then how many sets of three can we break six into? The answer is of course 2, because

\ 6 = 1+1+1+1+1+1 = (1+1+1)+(1+1+1) = 3 + 3 ; two sets of three.

Division Problems

Division is the first operation where a problem arises. In all the previously defined operations (addition, subtraction, and multiplication) we could perform the operation on any pair of numbers we chose. However, with division we cannot divide by zero. Much will be said about this fact throughout the course of this book, and even through your studies in all of mathematics.

Exponentiation[edit]

What are exponents?

Exponents are a shorthand used for repeated multiplication of the same number. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that: 4 × 5 = 5 + 5 + 5 + 5. The expression "4 × " told us how many times we needed to add. Exponents are the same type of shorthand for multiplication. Exponents are written in superscript (that is, a smaller number written above) after a regular-sized number.

For example: 23 = 2 × 2 × 2 = 8. The number in larger font is called the base. The number in superscript is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is called the base and 3 is called the exponent.

The expression 23 is read aloud as "2 raised to the third power", or simply "2 cubed".

Here are some other examples:

6 × 6 = 62 (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".)
7 × 7 × 7 × 7 = 74 (This would read aloud as "seven times seven times seven times seven equals seven raised to the fourth power". There are no alternate expression for raised to the fourth power. It is only the second and third powers that usually get abbreviated because they come up more often. When it is clear what is being talked about, people often drop the words "raised" and "power" and might simply say "seven to the fourth".)

There are two other powers that play an important role in understanding powers. One is

2^0

and the other is

2^1

Any number raised to the power of zero is equal to 1. For example: 6^0 = 1
Any number raised to the power of one is itself. For example: 15^1 = 15


In general, an exponent of a number to power of n

a × a × a ... = a^n

The base is a and it is multiplied by itself n times.

When we look at exponents again later in this book we will see that changing the type of number used for the exponent begins to have some very sophisticated results. To get a feel about why some people enjoy math read about mathematical beauty and the equation e^{i \pi} + 1 = 0\,\!.

Exponentiation Problems

Roots[edit]

What are Roots?

Roots are the opposite for exponents. It's easy, although perhaps long, to compute exponents given a root. For instance 7*7*7*7 = 49*49 = 2401. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? This article gives a formula for determining the answer, while this article gives a detailed explanation of roots.

Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When we graph functions we will see that polynomials that use exponentiation use curves instead of lines. Using algebra we will see that not all of these polynomials are functions, that knowing when a polynomial is a relationship or a function can allow us to make certain types of assumptions, and we can use these assumptions to build mental models for topics that would otherwise be impossible to understand.

For now we will deal with roots by turning them back into exponents.

The positive nth root of x is represented as \sqrt[n]{x} = r. We get rid of the root by raising our answer to the nth power, r^n = x.\!\,

Problems Using Roots

Logarithm[edit]

Logarithm is an operation that is similar to exponentiation and roots.