This section provides an introduction to numbers. It describes how they are created and what they mean or represent. In the process, the idea of counting is developed. The concept of place value is introduced, as is the concept of the number line.

#### 2.1 How We Make Words

To gain some insight into how we make numbers it may help the student first to understand how we make words.

All the words in the English language are created from a *set* of **symbols** that we call **letters** of the alphabet: {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}, {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z} and sometimes the symbols {‘,-}. Each letter has a lower **case** **representation** as shown in the first set above, and an upper case representation as shown in the second set above and sometimes called capital letters. There are rules about which case to use and when, as well as many other rules telling us how to make words from individual letters.

There are thousands of symbol used in different languages throughout the world as well as in mathematics. You can design your own personal symbol too (try it) but no one else would know what it meant. We note that only certain arrangements of letters have meaning as words. For example, the **string** of letters “fvxayob” is not a word in the English language. We could create our own secret language that would include this string as a new word, but it would be difficult to pronounce and no one would know what it meant.

Most letters of the alphabet have no meaning as symbols. The letter “c” in the word “cat” has no meaning although the entire word means a cute kind of furry animal. There are three letters, however, that do have meaning in addition to their use as symbols in making words. These are the single symbol words {a,I,O}. The word “a” has the meaning of “one” as in “a cat” meaning “one cat” while the letter “a” making the word “cat” has no meaning as a symbol in the word “cat”.

This can be a difficult point to grasp. We express the idea that the three letters {a,I,O}, have both representation as a symbol and meaning as a word, by the term “*overloaded**“*. Each has two uses, one as a letter for building larger words and one as a word in itself. We will see in the next section how numbers are made in a manner similar to words.

#### 2.2 How We Make Numbers

Like words, numbers have meaning. They usually represent a quantity of something such as four cats or a **value** such as five dollars. As we shall see shortly, every number is created from the set of symbols that we call **numerals** or **digits**: {0,1,2,3,4,5,6,7,8,9}, and in some cases in later lessons, using some special symbols: {+,-,.}. Whereas every letter is a unique symbol that can be combined to make words that have meaning, the ten numerals can be combined to make numbers with meaning.

Unlike letters of which only three have meaning by themselves, we shall see that *all* the digits have meaning by themselves (all are overloaded). In the next section we will use the idea of counting to see how we can make any number from just ten digits.

#### 2.3 Counting

If you are asked how many red marbles are in your pocket and you say there are none, we would like a symbol to express the idea of having none of something. For this we use the digit “0” which means a quantity of zero. If on the other hand, you have one or more red marbles, we figure out how many by **counting** them.

Zero seems to be a reasonable point to start counting from. Suppose I tell you that I have many pets and you want to know how many cats I have. To get the answer we play a guessing game with two rules:

- Rule 1: you have to start by asking me the smallest number of cats that I might have.
- Rule 2: if a guess is wrong, you have to count (i.e. “add”) one more cat and ask if I jave this new number of cats. You keep counting and asking in this way until your guess is correct. [Note to teachers: play this game with the student.]

For example, if you guessed four cats and I said no, you must use rule 2 again. You must add one more cat and guess five cats (you can use your fingers to help). Because I don’t have parts of cats running around the house, when you guessed four cats and then five, there was no number of cats in between such as four and a half cats. This is true for every guess you make by counting.

So in this game you counted every **whole number** (think of adding whole cats) until either we got tired or I ran out of cats. If we continue the game tomorrow we will count and create even larger numbers. If you wondered what kind of numbers we might need to create to count parts of things like pieces of pie, we will get to this in the later section on *Fractions*.

For alternate instruction with simple exercises see the Kahn Academy videos: *Counting*.

#### 2.4 Symbolic Representation of Numbers

According to rule 2, after guessing 0 cats you must ask me if I have one cat. For this we use the digit “1” to represent the quantity of one cat written as 1 cat.

When I answer “no, I do not have only 1 cat”, by rule 2 you must guess again. The rule allows you only one guess, the number arrived at by adding 1 cat to the last guess which was 1 cat. This number is two cats. We represent the quantity two by the digit “2”.

Again I answer “no”. You keep counting, increasing the number you guess by 1 cat each time. Finally you guess nine cats and represent this number by the digit “9”. However, again I say “no”.

Now things get tricky. Your next guess is 1 more than 9 and that we call ten. You have no more distinct, unused digits left to represent ten. Then, remembering that words can be made with more than one letter, you decide to try and make larger numbers by using more than one digit.

First, however we write our first *group of 10 numbers* (quantities – remember that 0 counts as a number) in a straight line:

When making words, we start from the leftmost position which is the front of the word and add letters on the right working towards the end. With numbers we do the opposite. We start at the rightmost end position of the number and add digits working towards the front of the number on the left.

We like to be neat and orderly so our moms won’t yell at us. So when adding digits to make larger numbers we select them from the set of digits, not in any old order, but in the same order as the first 10 digits we used to make single digit numbers.

If you wonder why we didn’t start with the digit “0” in front of the single digit numbers in the row above, we will explain why in a moment after you learn a bit more about what a number’s representation means.

Now we remember that in counting cats, the number of the first cat you counted starting from 0 was made with a “1”. To keep the same order of digits for your two digit numbers we do this by placing a “1” in front of each of the single digit numbers.

The first of the new larger numbers is “10” representing the quantity ten and written as 10. Using the game rules, the number that is 1 larger than 10 we decide to call eleven and represent by the digits “11”. This is the first group of 10 numbers and 1 more. Ten fingers and one toe if you like. I have many cats and using rule 2, you get up to the number of 19 cats represented by the digits in order as “19”. We will place these 10 new numbers in a line below the first line:

0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |

10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |

The number 3 in the first line or group of 10 numbers has 3 numbers in the line before it (o, 1, 2 cats). Immediately below 3 in the second line above we have the number 13. It has 13 numbers before it, 10 in the first line or group of 10 plus 3 in the second line. Although 13 is in the second line or group of 10, there are still more numbers we can make in the group so we don’t use a “2” in front of the single digits until this new group of 10 is full when we reach the number 19.

At this point we have used up all 10 digits available to us to use with the digit “1” in front. We now know how to solve this problem by using the digit “2” in place of “1”. The next new number will be “20” representing the quantity 20 or 2 groups of 10 that have already been counted, and written as 20. From 20 we count more cats in the same way as we counted more cats than 10. This creates a third row in our table:

20 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |

By now you must realize I have a lot of cats, but you are determined to find out how many so we keep playing the game. It continues by using rule 2 until we count 99 cats. We write all the numbers created so far, in the table below:

00 |
01 |
02 |
03 |
04 |
05 |
06 |
07 |
08 |
09 |

10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |

20 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |

30 |
31 |
32 |
33 |
34 |
35 |
36 |
37 |
38 |
39 |

40 |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
48 |
49 |

50 |
51 |
52 |
53 |
54 |
55 |
56 |
57 |
58 |
59 |

60 |
61 |
62 |
63 |
64 |
65 |
66 |
67 |
68 |
69 |

70 |
71 |
72 |
73 |
74 |
75 |
76 |
77 |
78 |
89 |

80 |
81 |
82 |
83 |
84 |
85 |
86 |
87 |
88 |
89 |

90 |
91 |
92 |
93 |
94 |
95 |
96 |
97 |
98 |
99 |

100 |
101 |
102 |
103 |
104 |
105 |
106 |
107 |
108 |
109 |

We have changed the first row in the table, the purple numbers, by placing the digit “0” in front of the single digits. We spoke of this earlier. Now we can see that the number 03 means that it counts the cats in 0 groups of 10 which is no cats, plus 3 gueses when making the number 3.

When we made the number 13, the “1” meant that we had already counted 1 group or line of 10 numbers. If we place the digit “0” in front of the number 3, it means that we have counted 0 or no groups of 10 before it so that o3 counts the same quantity as 3. In the next section on *Place Values* it will become more apparent why we might want to do this.

You will have noticed the last line of blue numbers. The next number after 99 has to be made with another digit added in front since we have run out of unique second digits to add. This is the number one hundred made from the digits “100” and representing the quantity 100.

You may have noticed an interesting property of this table. using the 10 original single digit numbers. By adding a “1” in front of each we get 10 new numbers. Since we have 10 digits to place in front of the single digits (remember 0 is a number), we get 10 times 10 or 100 unique numbers. With a third digit added in front of each 2 digit number we can create a total of 10 times 100 or 1000 unique numbers (We have used multiplication to get the total numbers. You will learn how to do this soon enough.)

So you have now guessed I have up to 100 cats. I tell you that there is no limit to the number of digits you can add at the front of a number. But I warn you, I still have many more cats than this which I now must go and feed.

#### 2.5 Place Value

When we learned to make numbers in the last section, we started at the right-hand end of each number and added positions or places as necessary to the front of the growing number. This first position has a name or **place value** called the **units** or “ones” place. The next place to the left of units is called the *tens* place, because this is the place that counts groups of 10. To the left of the tens place is the **hundreds** place which counts groups of 100 and then the thousands place and so on. How many places in a number do you think you need to make one zillion? Lots probably.

Although we build numbers from right to left, we read them by their place values from left to right as with reading words. So the number is 321 is read as three hundred (hundreds place) and twenty (tens place) one (units place).

#### 2.6 The Number Line

We will find it useful to build what we call a **number line**. Follow these steps:

- Take a ruler and lay it on a piece of paper;
- draw a line starting from the left-hand end of the ruler to either the right-hand end of the ruler if the paper is big enough or to the edge of the paper;
- without moving the ruler, make a mark on your line next to each of the bigger lines on your ruler that have a number beside them;
- mark the start of the line and place a 0 above it – for some reason, rulers never label the 0 mark or start;
- place the numbers that are on the ruler, next to the marks on your line; and
- note that between numbered marks on your ruler there are many evenly spaced marks in between numbers.

Because your paper is small you can only get a few numbers on it, the numbers you made by counting and starting at zero. Imagine if you took a roll of toilet paper and used an extremely long ruler to create a very long number line. You can imagine that the last number on the long piece of toilet paper might be 100 or even more. But just as we still had more cats when you were counting them, there are more numbers to place on the line no matter how long a line you can draw. (Warning: do not try the toilet paper exercise at home. Your parents will not be impressed!)

#### 2.7 Key Points

In the lists below, any text in red is mostly for information purposes. Although words and ideas in red may be used in more than one place in this course, the student may ignore them as nor part of the core ideas or for more advanced mathematics..

##### 2.7.1 New Words

**case**: for the English language, every letter has two representations. One is called lower case as in the letter “a” and one is called upper case as in the letter “A”;
**counting**: this is the process whereby we identify or mark objects by a sequence of numbers that increase by a fixed amount. We usually count by ones as we did in counting cats but we might count something like shoes in twos;
**digit**: a symbol that is used to represent a number or is used with other digits to make a larger number; also the same as *numeral*;
**ellipsis**: the three dots in brackets, {…} meaning “and so on”;
**hundreds**: the third place (from the right) in a number;
**leading zero**: a zero in the leftmost place of a number that does not change its value and is therefore seldom written;
**letter**: a symbol from the alphabet that is used with other letters to make a word;
**number line**: an imaginary line with equally spaced, numbered marks for all the numbers that you can make up;
**numeral**: another term for *digit*;
**ones**: the first place (from the right) in a number; also, *units*;
**overloading**: a technical term for a symbol that has two or more different uses;
**represent**: to stand for something, for example, the string of symbols “cat” stands for or represents a cute furry little animal;
**place**: the position in a number where a digit is found;
**place value**: the name of the position of a digit in a number counting from the right;
**set**: a collection of objects that are listed in curly brackets: “{…}”;
**string**: a row of symbols that are joined together in a distinct group. Words are strings of letters and numbers are strings of digits, something like a “string” of beads;
**symbol**: a mark or character that is used to construct words or numbers;
**tens**: the second place value (from the right) in a number;
**units**: the sames meaning as *ones*;
**value**: the meaning of a number, usually a quantity that the number represents or counts; and
**whole numbers**: numbers that count whole quantities and not part quantities.

##### 2.7.2 New Symbols:

**digits**: {0,1,2,3,4,5,6,7,8,9}; the set of symbols from which all numbers are made;
**ellipsis**: “…” meaning “and so on”; and
**letters**: {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}, {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z} and {‘,-}.

##### 2.7.3 New Rules

- Build a number from right to left using only one of the 10 digits in each place;
- Create all the numbers that we have used by counting by ones or adding 1 to the last number created. Pay careful attention to all the place values (adding 1 to 99999 creates the number 100000).

##### 2.7.4 Conventional Practices

- To make writing and reading large numbers easier we separate place values in groups of 3 by using a comma, “,”. In rule 2, 99999 could be written as 99,999. The comma, however, is never used in mathematical expressions or calculations – it’s for appearances only.
- A number may have the digit “0” added to its front without changing its value. For example 645 and 0645 represent the same quantity. Called a leading zero, it is never written as part of the number.