Category Archives: Math

Introduction to Arithmetic Operators

Recall that we have described arithmetic as a game. In the last section we studied numbers and these numbers are the game pieces

What Is an Operator?

Consider this definition of a (telephone or switchboard) operator from the Merriam-Webster dictionary: a person whose job is to help to connect telephone calls. More generally we think of an operator as a person who foes some kind of work. In mathematics we have many operators but not of the people kind. You will meet four in this section and learn a bit about the work they do.

The operators used  in arithmetic are addition or “plus“, subtraction or “minus“, multiplication and division. We’re mathematicians and don’t like writing so we invent symbols to represent our operators. The symbols for addition and subtraction are {+,-}. An example of 3 plus 2 would be 3+2 and an example of 3 minus 2 would be 3-2.

The symbols for multiplication and division, {x,÷}, cause us problems, however.  The “x” would be OK for multiplication in arithmetic but won’t work for algebra that you will learn soon enough, because in algebra we use “x” to represent an unknown quantity.

The Kahn Academy uses a special kind of dot, “•”, not to be confused with a period “.”. But the “•” symbol does not appear on a computer keyboard. Instead it has to be looked up in a special table, something that is time consuming and may not be possible for you. Computer programmers who type all day use the asterisk, “*”, as a multiplication symbol and that is what we will use. An example of 3 multiplied by 2 would be 3*2.

Division presents even more problems. The standard symbol used as the division operator is “÷”.  Like the dot this is not on your keyboard. Text books represent long division by a symbol that looks like this: “)¯¯¯¯”. We can’t even fully express a problem with this symbol. The symbol that is on the keyboard that we will use for division is the forward slash, “/”. An example of 3 divided 2 would be 3/2.

Properties of Operators

Operators have a special word that identifies them, the word “operator”. Like the switchborad operator, the arithmetic operators do

Arithmetic Expressions

An operator is a person who does some kind of work, often between two people. If your computer breaks and you phone the company for help you may reach a receptionist or operator who will connect you to the right person to speak to.

In arithmetic our 4 operators do work that applies one number to another. The statement for the operation of subtraction that we wrote about above was 3-2. Such statements are called expressions. We will use this term a lot.

Key Points

In the lists below, ant text in red is for informational purposes only and the student need not remember them in this course.

New Words
  • digit: a
New Symbols:
  • digi
New Rules
  • buil

An Introduction to Algebra

  1. Equations
    1. The use of Variables
    2. Solving One Equation in One Unknown
    3. Key Points
      1. New Words
      2. New Symbols
      3. New Rules
      4. Conventional Practices

Negative Numbers

What Are Negative Numbers?

Today I was watching the weather report on TV. It said the high for today will be 0 degrees Celsius. It also told me that the high yesterday was 6 degrees above zero or+6 and that tomorrow the high will only be 6 degrees below zero or -6. You are probably familiar with such reports and may even have a thermometer outside that you can read yourself.

But why did the weatherman on TV say plus 6 and minus 6 instead of just saying 6? The high yesterday was just as many degrees away from zero by 6 degrees as the high tomorrow will be away from zero by 6 degrees. We can’t just say the value of the distance from zero but we have also to say the direction away from zero, plus or minus, or with symbols, + or -. Numbers with a sign of + are called positive numbers and numbers with a sign of – are called negative numbers.

Until now, numbers for us have been adequately represented by their value. Now we have to add a new property of numbers and that is called sign. We also now want to be a bit more precise and qualify the value of a number as its absolute value. The idea of absolute value is important in computer programming and higher mathematics but not so much for us now.

A Few Things About Signs

When used to denote the sign of a number the symbols {+.-} are acting as operators. They have only one operand however, and are therefor called unary operators.

Further we have another unary operator,, the absolute value symbol, “| “, that operates on a number to give just its value without any sign. An example is the expression | -6= 6.

You may have noticed that all the numbers we have used until now have shown no sign. Since most numbers we use are positive numbers, by convention, that means by agreement by most mathematicians, we don’t write the + sign for positive numbers. After all, we are mathematicians and don’t like extra writing. This causes a bit of possible confusion when we use the absolute value operator. We have to remember that the expression above is meant as it is written and does not stand for | -6= +6.

Updating the Number Line

Addition and Subtraction of Negative Integers

Multiplication and Division of Negative Integers


What Are Integers?

The numbers that we made in Introduction to Numbers are called integers or whole numbers. In that section we counted our cats. The numbers we made counted whole cats since we don’t have any partial cats – say a half of a cat – walking around. We also have pies in our house, but someone keeps eating pieces of the pies. We can’t count our pies because we do have half pies in the cupboard. We therefore need some way to count parts of pies and we will explore these ways later in Fractions and

The Integers (Whole Numbers) and the Number Line

  1. counting cats and pies.
  2. The numbers we have talked about are the numbers {0,1,2,…,10,11,12,…,20,21,22,…} these are called integers or whole numbers. Supposing you had half a pie left from supper. What kind of a number would a mathematician use to represent half a pie? We will find out later, but if our integers represent pies, the number to represent half a pie isn’t among them. Hence the integers represent the numbers of whole pies or whole numbers.
  3. Take a ruler marked in centimeters, draw a line from one end of the ruler on the left  to the other and mark on it the integers shown up to 10.
  4. The first thing you may notice is that there is no number at the beginning of the ruler which is the first mark on the ruler. This is actually the number 0.
  5. The next thing you may notice is that between any two numbers on the ruler there are 9 marks in between. We will talk about their meaning later.
  6. Depending on the length of the ruler, the line you draw may allow up t0 30 marks that are numbered and evenly spaced. Longer rulers would allow us to mark more numbers on a longer number line. To be able to mark 1,000 on the number line, we would need a ruler possibly longer than the room you are in. Another characteristic of mathematicians I forgot to tell you is that big numbers don’t scare them. In fact some get excited over the size of really, really big numbers that they invent.

Addition and Subtraction of Integers

Multiplication and Division of Integers

Comparison of Integers

Key Points

A number has the following properties:

  1. It has a representation consisting of the ten digits.
  2. It represents a quantity or value.
  3. The position of each digit in the number is a place value with a distinct name.

1 Preamble for Instructors and Self-Learners

This is a course intended as an introduction to numbers and arithmetic. It attempts to make the concepts understandable to persons of young age and/or little background in numeracy. The course was motivated partly by our work with our grandson and partly by the appalling sparseness of the elementary school mathematics curriculum in Ontario.

2 Introduction to Numbers

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.

Multiplication of Integers

Decimal Multiplication

The Decimal Point

In Section 3, we discussed numbers called integers, that could represent a whole quantity such as 3 pies. But how can we represent just part of a whole? What kind of a number would represent half a pie?

Since numbers are built up with place values from 1 to 10from 10s

An Introduction to Numbers and Arithmetic

This page is the start of an introductory course in numbers and arithmetic. We take the overview that all mathematics is a collection of games. Each field of mathematics can be reduced to a set of symbols or game pieces, and a set of operations on the symbols or game rules. The fun and the challenge is to see what new outcome can be found by exploring the interaction of the game’s pieces according to the rules of the game.

It provides part of an elementary curriculum in mathematics. Below is the table of contents.

Under heavy construction.

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