Algebra/Chapter 1/The Number Line

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Algebra Chapter 1: Elementary Arithmetic Section 1: The Number Line

Operations of Arithmetic

1.1: The Number Line

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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

Numbers are 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.

The Ishango bone on exhibition at the Royal Belgian Institute of Natural Sciences

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 Number Line

A 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, neative 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".

Symbol	Explanation	Example
=	When two values are equal, we use the "=" symbol.	2 + 3 = 5
>	When one value is greater than another, we use the ">" symbol.	40 > 39
<	When one value is less than another, we use the "<" symbol.	14 < 17

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.

Types of Numbers

Whole Numbers

All numbers consist of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 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.

Example 1.1: How would you write the number "three-thousand five-hundred twenty-two"? 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.

Caution! 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

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.

History Of Zero 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.

Practice Problems

Conceptual Questions

Problem 1.1 (What is a Number?) Define what a "number" is in your own words.

Problem 1.2 (Difference of Decimals) What is the difference between "ten" and "one-tenth"?

Problem 1.3 (Picture Perfect) Suppose the number line actually existed physically. Would you be able to take a photo of the entire number line if you backed away far enough?

Problem 1.4 (Explaining the Writing of Numbers) Explain in your own words how you write numbers, both in word form and with numerical symbols.

Problem 1.5 (Largest Number Possible) What is the largest and smallest three-digit number you can write using the digits 0, 8, and 4? Use each digit only once, and explain how you obtained your results. If you wrote these numbers to the right of a decimal point, what is the largest number you can make.

Problem 1.6 (A Million) A million is one thousand thousands. Explain how this is so.

Problem 1.7 (Reading it Wrong) Explain what is wrong with reading "50,002" as "fifty-thousand and two". Explain what is wrong with reading "2.203" as "two and two hundred and three thousanths".

Problem 1.8 (Number Associations) What whole numbers are associated with each word?

a. zilch
b. duo
c. decade
d. a pair
e. naught
f. trio
g. four score
h. century

Problem 1.9 (Problem with Fractions) Why can't we say that 3/5 of the figure below have been shaded in?

Image: 300 pixels

Problem 1.10 (Large Numbers) Determine if the following is true: "The more digits a number has, the larger it is".

Problem 1.11 (Signs) A fast-food menu has the cost of a hamburger listed as .99¢. Explain what is wrong with this.

Exercises

Problem 1.12 (Locating Numbers) Copy the number line below, and then figure out where the following values might be located on it.

${\frac {3}{11}},0,0.0001,5$

Problem 1.13 (Comparing Numbers) For each given pair of numbers, determine which of the two is larger.

Problem 1.14 (Weighing Bull Sharks) A biologist is studying bull shark populations. She records the weights of four sharks, in pounds, that she has caught. Order the bull sharks from lightest to heaviest.

Problem 1.15 (Place Values) Find the place value of the number 5 in each of the following numbers.

Problem 1.16 (Writing Numbers) Translate to mathematical symbols

Problem 1.17 (Writing Numbers in Words) Write the following numbers in words

a. 9
b. 10
c. 274
d. 8,322
e. 1,000,000,009
f. 1,343,234,985
g. 0.01

Problem 1.18 (Numbers in Expanded Form) In the number 7,893, there are "7 thousands", "8 hundreds", "9 tens", and "3 ones". We therefore say that a number is in expanded form when it is written as follows:

7 thousands + 8 hundreds + 9 tens + 3 ones
or
7000 + 800 + 9 + 3

Write the following numbers in expanded form:

a. 473
b. 6852
c. 73,016
d. 570,003
e. 3,519,803
f. 48,000,061
g. 37.89
h. 124.575
i. 7496.5467
j. 6.40941

Problem 1.19 (Fraction Diagrams) Write a fraction to describe what part of the diagrams below are shaded. Write a fraction to describe what part of the diagrams aren't shaded in.

Problem 1.20 (Fruit Basket) A basket of fruit holds 5 mangoes, 7 apples, 12 oranges, and 20 pomegranates.
a. What fraction of the fruits in the basket are apples?
b. What fraction of the fruits in the basket are not oranges?
c. What fraction of the fruits in the basket are oranges or pomegranates?

Reason and Apply

Problem 1.21 (Count the 24ths) How many ${\frac {1}{24}}$'s are in ${\frac {2}{3}}?$

Problem 1.22 (Negative Negative Negative Negative...)

a. What is $-(-(-2))$?
b. What is $-(-(-(-2)))$?
c. What if there were 20 minus signs in front of the 2?
d. What if there were 75 minus signs in front of the 2?

Problem 1.23 (Using Bar Graphs) Look at the diagram below, and use it to answer the following questions.

Problem 1.24 (Using Multibar Graphs) Look at the diagram below, and use it to answer the following questions.

Problem 1.25 (Using Line Graphs) Look at the diagram below, and use it to answer the following questions.

Problem 1.26 (Creating a Bar Graph) Look at the table below, and use it to create a bar graph.

Problem 1.27 (Reading Meters) The amount of electricity in a household is measured in kilowatt-hours. Determine the reading on the meter shown below. (When a pointer is between two numbers, use the smaller number).

Problem 1.28 (Sky High) The table below shows the altitude each of the cloud types are found at. Graph the numbers on the vertical number line below.

Problem 1.29 (Rulers) Look at the diagram of a ruler below.

a. How many tick marks are between 0 and 1?
b. What number is the arrow pointing to?