Algebra/Chapter 1/The Number Line

1.1: The Number Line

The concept of number is the basis of all of modern mathematics. Numbers have many uses, mostly for counting and measuring things. This section will cover the concept of numbers, as well as the different types of numbers that are seen in mathematics, and their uses.

An Introduction to Numbers
Many of us take the concept of "number" for granted.

Numbers are an abstract concept that often used to count or measure things. Though, depending on where they're used, they may be used for many different purposes. Numbers are merely an idea in our head, things that we talk about and write about, but never actually see beyond that. To do this, we represent these numbers using symbols called numerals, such as "5" or "five". We could also represent the number by holding up 5 fingers, tap a table 5 times, or jump and down 5 times. These are only a few ways to represent the number 5.

Generally speaking, a number is an idea, and a numeral is a way that we can express that idea with. Numbers represent what is known as a quantity. This is how much of something there is in a group of objects, such as “six” lollipops, “seven” slices of pizza, “twenty-five” dollars, or “one hundred” millenniums. To represent these quantities, we can say 6 lollipops, 7 slices of pizza, 25 dollars, or 100 millenniums.

Number Systems
Numbers can be denoted in many ways. The collection of symbols and notation that we may use to represent these numbers is referred to as a number system. Number systems make use of digits or other symbols in a consistent manner. For example, in the earliest of civilizations, people had been believed to rely on the counting of their fingers, or a group of sticks and pebbles. It is not known for certain what the earliest number system may have been, but quite early, groups of symbols, |, ||, |||, ||||, etc. were used to represent numbers. These are still used today as tally marks, |, ||, |||, ||||, and |||| as 1, 2, 3, 4, and 5 respectively.

One of the oldest, and most useful, applications of number is time. We use time in order to keep track of the order of a sequence of events, to compare the duration of events or the intervals between them, as well as to quantify the rates at which things change. There have been numerous ways in order to keep track of time, from the prehistoric tribes who kept track of time through the rise and setting of the sun, to the markings on bones and trees.

During the late Stone Age, hunting and fishing societies had developed in the Nile Valley and in many parts of Africa. An interesting find from these ancient civilizations is a carved bone discovered in Lake Edward in the Democratic Republic of Congo. This bone, referred to as the Ishango Bone, had several notches in it, indicating a pattern. The exact purposes of this bone are unknown, but from a mathematical perspective, this may be viewed as a sort of tally system.

The bone has several separate markings. One set of markings has four groups of 11, 13, 17, and 19; another 11, 21, 19, 9; and the third has seven groups of 3, 6, 4, 8, 10, 5, 5.

The Hindu-Arabic Numeration System
The number system that we use today is referred to as the Hindu Arabic numeration system, which, as implied by its name, were created by the Hindus before the third century, this number system was introduced into the Western World 1000 years later by Italian mathematician Lenoardo Fibonacci. This number system would then be popularized by the Arabs, hence its name.

All numbers in this system consist of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Numbers are written using these digits, placing them in certain positions. The Hindu mathematician who devised the system had stated that "from place to place each is ten times the preceding." For instance, the number three is written with the symbol "3", and the number two million is written using the numeral "2", followed by six zeroes.

Place Values
As you may know, we arrange the digits in a number by their place value. When numbers are composed of more than three digits, we sometimes use commas to separate the digits into three. These groups of three digits are referred to as periods. In the Hindu-Arabic number system, the positional value in the first period are the ones, tens, and hundreds.

Every other period in a number also consist of a one, ten, and hundred. When we read the numbers from right to left, notice that the value of each position is ten times the preceding one.

Order of Numbers
The order of numbers is the method we use in order to arrange numbers from "smallest" to "largest".

The Number Line
The number line is a line that shows all numbers, with the left side decreasing, and the right side increasing. The typical number line shows all integers (whole numbers), but it can consist of the numbers between 2 numbers, like half or 3.7.

As one may gather from the above, numbers come in many varieties. For now, the numbers we shall be discussing about are whole numbers, decimals, fractions, negative numbers, and zero.

The magnitude of a number is its distance from zero on the number line.

For the rest of the section, we will be covering all of the features of this representation of numbers one at a time.

Comparing Numbers
The dictionary definition of "comparing" two things is viewing them in relation to one another. For this course, it is good to be able to compare numbers, or be able to know when one quantity is equal to, less than, or equal to another. In mathematics, the symbols that we use for comparing numbers are ">", which means "greater than", "<", which means "less than", and "=", which means "equal to".

In the diagram below, we see that 7 is greater than 3 by 4. In simple terms, this can be expressed as "7-3 = 4".

In case you have forgetten which number goes where, just remember the following:

BIG > small small < BIG

The small end will always point to the smaller number, and the larger end will always point to the larger number.

The number line can also be utilized as a tool in order to compare numbers, and it is very easy. All that one needs to know in order to do this are two simple facts:


 * For any two numbers on the number line, the number farther right is greater.
 * For any two numbers on the number line, the number farther left is smaller.

Whole Numbers
The most familar number system that is used is known as the counting numbers, more technically refered to as the natural numbers. These are the numbers 1, 2, 3, 4, and so on. These numbers are either the numbers you and I used for learning how to count, as indicated in the name, (as in "there are six pencils on the table"), as well as for ordering things (as in "this is the fourth smallest ball in the basket"). For large numbers, commas are used to separate digits into groups of 3.

In this example, the digit 3 is in the thousands place, the digit 5 is in the hundreds place, 2 is in the tens place, and 2 is in the ones place.

$$3,522$$

Fractions and Decimals
We understand how to count whole numbers. But what happens when we are not working with whole parts? Most of the time, you will come across instance "uncountable" objects, or things that cannot be represented as a group of objects.

Whereas whole numbers are useful for counting things in a group, in most instances, we are interested in measuring things that do not have quantities as precise as "one" or "three-hundred" or even "one million". We can "count" things like holes in a piece of cloth, the number of slabs of wood used to make a fence, and the number of stones in a lake. We cannot however "count" things like the amount of water in a bucket or the weight of an object. Instead, special tools are used to determine these quantities, such as rulers for length, scales for weight, and stopwatches for time.

Decimals can be thought of as the numbers that are found between the tick marks found on the number line. These are often identified with a decimal separator symbol, typically ".", which is placed between the digits (Though some countries use a comma "," instead). The digits on the right of this symbol represents the part that is between the tick marks. For example, the number 12.5 is between the numbers 12 and 13. It is more than 12 but less than 13. The number 12.3 is also more than 12, but it is also less than 12.5.

Fractions are also numbers that can be found between the tick marks on the number line. They also represent the idea as decimals, values that include part of a whole, but they are a different way to express that idea.

Contrary to how you might have been taught, you do not pronounce a "," in a number as "and".

It should also be noted that in some countries, the "," and the "." switch places. So "1,523.99" in America might be written as "1.523,99" elsewhere.

Negative Numbers and Zero
Throughout the history of mathematics, there were certain ideas that even the greatest scholars had trouble accepting that we now take for granted today, some of which today's youth can easily grasp. Two prime examples of this were the concept of "negative" numbers, and the number zero.

Mathematic's origins were routed in counting things, like the number of sheep one owned. If you had two sheep, you'd count them as "one, two". Logically, that meant that one could not ask you to "lend them three sheep", since you'd have no more sheep to give up after the two you already had.

Negative Numbers are the numbers found on the left of the number line. In other words, these are the numbers that are less than zero. Imagining a number less than zero, can be difficult. If you have a plate of cookies, and everyone takes all of them until there are none less, it is hard to imagine being able to take any more cookies.

Zero is the value which denotes that there is no amount, for example, if a person has zero hats, that means that they have no hats. Zero is found between the positive and negative numbers, located halfway between -1 and 1. The number zero is neither positive of negative, but it is even.

Long ago, the concept of zero was non-existent. However, the concept was first brought about by the Babylonians, Ancient Indians, and several Central American tribes at different times, though the number primarily served as a placeholder rather than an actual number. Zero as a number is thought to have been introduced in India as far back as 628 AD in the Brāhmasphuṭasiddhānta, which was theorized by the mathematician Brahmagupta.

However, some countries still did not impliment the number zero, which meant that mathematics was significantly more difficult to do.