QMI101
Component 1
Study Unit 1.1 - Priorities and laws of operation
Commutative, Associative, Distributive Laws
Why does (3 + 4) + 10 give you the same answer as (4 + 3 ) + 10 ?
5(2 + 6) will produce the same result as 5(2) + 5(6), but why?
There are many laws which govern the order in which you perform operations in arithmetic and
in algebra. The three most widely discussed are the Commutative, Associative, and Distributive
Laws.
The Commutative Law
( "change" the order of the numbers or letters)
Over the years, people have found that when we add or multiply, the order of the numbers will
not affect the outcome.
5 + 4 is 9 and 4 + 5 is 9.
The 5 and the 4 can be switched in addition.
9(2) is 18 and 2(9) is 18.
The 9 and the 2 can be switched in multiplication.
"Switching" or "changing" the order of numbers is called "commuting".
People "commute" from home to work everyday. You travel from where you live to where you
work, and at the end of the day turn around to return to where you live.
When we change the order of the numbers, we have applied the "Commutative Law".
In an addition problem, it is referred to as the "Commutative Law of Addition".
In multiplication, it is the "Commutative Law of Multiplication."
The Commutative Law does not work for either subtraction or division.
The order of the numbers will affect the outcome.
When you write 100 - 25, it is very different from 25 - 100.
The first case could be likened to writing a R25 check when you have R100 in the bank.
The other case, 25 - 100, would be an example of writing a R100 check when you have R25 in
the bank. Oops.....
One must be cautious with subtraction. If we look at the operation of subtraction as "adding the
opposite" then there may be a way to "commute" the numbers, but in terms of addition and not
subtraction.
5 - 1 is not the same as 1 - 5.
The first case produces positive 4, but the second case yields negative 4.
We can not commute the numbers in subtraction.
Yet, if we think of 5 - 1 as 5 + (-1), five plus the opposite of one, then the operation has been
changed from subtraction to addition. And we can commute in addition.
5 + (-1) is the same as (-1) + 5. Both "additions" will produce the number positive 4.
KEY IDEA:
We can commute when we add or multiply, but we can not commute or "change" the order of the
numbers when we divide or subtract. So (3 + 4) + 10 will give you the same answer as (4 + 3 ) +
10 because we used the Commutative Law of Addition.
The Associative Law
( "the parentheses shift and the numbers do not ").
We have learned to add two numbers at a time, but when we have three or more numbers to add,
where do we begin? Does it matter?
This is why the Associative Law was created.
For example, consider 3 + 10 + 2
You could first add 3 and 10 to get 13. Then add the result to 2 and obtain 15.
( 3 + 10 ) + 2 = ( 13 ) + 2 = 15
Or you could first add 10 and 2 to get 12. Then add the result to 3 to get 15.
3 + ( 10 + 2 ) = 3 + ( 12 ) = 15
In both cases, we obtained the same answer.
( 3 + 10 ) + 2 = 3 + ( 10 + 2 )
Notice that the numbers: 3, 10, and 2 did not move.
What DID move was the parentheses.
In the first case, the parentheses were associated with the first two numbers 3 and 10.
The second time we tried the problem, they were placed around ( associated with) the 10 and 2.
The Associative Law allows you to move parentheses as long as the numbers do not move.
As with the commutative law, this will work only for addition and multiplication. The
Associative Law is similar to someone moving among a group of people associating with two
different people at a time. You talk to Will and Susan for awhile, then move your attention to
Susan and Bob. Bob is next to Anita, so you chat with the two of them. The people remain
standing in the same place as you turn your head to converse with different people. You
associate your attention to two people at a time, but can move your attention and not move the
people.
Examples of the Associative Law of Addition:
Look at both sides of the equation in the first step.
Parentheses move, but the numbers stay in the same order.
3 + ( 7 + 8 ) = ( 3 + 7) + 8
3 + ( 15 ) = ( 10 ) + 8
18 = 18
( 3x + 2x ) + 5x = 3x + ( 2x + 5x )
( 5x ) + 5x = 3x + ( 7x )
10x = 10x
Examples of the Associative Law of Multiplication:
5 * ( 5 * 6 ) = ( 5 * 5 ) * 6
5 * ( 30 ) = ( 25 ) * 6
150 = 150
2x * ( 3y * 4z ) = ( 2x * 3y ) * 4z
2x * ( 12yz ) = ( 6xy ) * 4z
24xyz = 24xyz
Why doesn't it work with subtraction?
3 - ( 5 - 2 ) = ? ( 3 - 5 ) - 2
3 - ( 3 ) = ? ( -2 ) - 2
0 = ? -4
because it does not produce a correct answer.
How about division?
15 / ( 10 / 2 ) = ? ( 15 / 10 ) / 2
15 / ( 5 ) = ? ( 3 / 2 ) / 2
3 = ? 3/4
Nope. It does not work with division.
KEY IDEA:
In the Associative Law, the parentheses move but the numbers or letters do not. The Associative
Law works when we add or multiply. It does NOT work when we multiply or divide.
The Distributive Law
("multiply everything inside parentheses by what is outside it").
Think of a delivery truck. It must move from the warehouse to several distributors along its route
unloading its merchandise at each business. When the truck has unloaded at one stop, it moves to
the next stop, unloads, and moves on until all locations have been visited. The distributive law
is somewhat like a delivery truck, it is distributing multiplication among terms. The truck is
outside the parentheses and the businesses are inside separated by plus and minus signs.
When we multiply two numbers, each of the numbers is called a factor.
When ( 5 ) and ( 2 ) are multiplied producing 10, the 5 is one factor and the 2 is another factor.
Now when we multiply ( 5 ) * ( 6 + 2 - 4 ) the 5 is one factor, but the other factor is an addition
and subtraction problem: 6 + 2 - 4. The 6, 2, and 4 are not factors. They are joined together with
addition signs and a subtraction sign making them "terms".
We've been told in other math classes to "do inside parentheses first."
Thus in the last example,
( 5 ) * ( 6 + 2 - 4 )
= ( 5 ) * ( 8 - 4 )
= ( 5 ) * ( 4)
= 20
But if we look at the problem in a different light, i.e. multiply each number inside the
parentheses by the number outside we would obtain:
( 5 ) * ( 6 + 2 - 4 )
= ( 5 ) * 6 + ( 5 ) * 2 - ( 5 ) * 4
= 30 + 10 - 20
= 40 - 20
= 20
The result is the same as above.
This is the idea of the distributive law. When you have parentheses in which there is addition
and/or subtraction, and when there is a factor outside of the parentheses, the factor may be
distributed to all terms inside the parentheses. Remember "terms" are separated by addition or
subtraction signs. In short, multiply every term inside the parentheses by the factor outside it.
Examples of the distributive law:
9 * ( 2 - 7 + 8 )
= 9 * 2 - 9 * 7 + 9 * 8
= 18 - 63 + 72
= -45 + 72
= 27
checking the answer: 9 * ( 2 - 7 + 8 ) = 9 * (3) = 27
5x ( 3x2 + 2x - 4)
= 5x (3x2) + 5x (2x) - 5x(4)
= 15x3 + 10x2 - 20x
How to check an algebraic problem.
In this example, the terms inside the parentheses are not alike. We can't add them.
The only way we could check our work would be to make-up a number for x and see if we
obtained the same answer.
Here's how:
Let x = 2 ( I just made it up, you could try your own number).
In 5x ( 3x2 + 2x - 4) replace the x by 2,
5*2 ( 3* 22 + 2*2 - 4)
= 10 ( 3 * 4 + 4 - 4)
= 10 ( 12 + 4 - 4)
= 10 ( 16 - 4 )
= 10 ( 12 )
= 120
Next, try the same number in 15x3 + 10x2 - 20x
15 * 23 + 10 * 22 - 20 * 2
= 15 * 8 + 10 * 4 - 20 * 2
= 120 + 40 - 40
= 160 - 40
= 120
The answers match, so it appears that the distributive law was done correctly.
KEY IDEA
The distributive law involves a number or variable outside of parentheses ( a factor ) and
numbers or variables inside parentheses separated by addition and/or subtraction signs ( terms ).
Multiply every term inside the parentheses by the factor outside it.
Thanks to the distributive 5(2 + 6) will produce the same result as 5(2) + 5(6).
Order of Operations
Please ---- Parenthesis or grouping symbols
Excuse ---- Exponents (and radicals)
My Dear ---- Multiplication/Division left to right
Aunt Sally ---- Addition/Subtraction left to right
Study unit 1.2 - Variables
A variable is a symbol that represents a number. Usually we use letters such as n, t, or x for variables. For example, we might say that s stands for the side-length of a square. We now treat s as if it were a number we could use. The perimeter of the square is given by 4 × s. The area of the square is given by s × s. When working with variables, it can be helpful to use a letter that will remind you of what the variable stands for: let n be the number of people in a movie theater; let t be the time it takes to travel somewhere; let d be the distance from my house to the park.
Variables are the thing that make algebra different from other kinds of mathematics like, say, arithmetic.
In arithmetic, we deal with numbers, and each number has a value, which is itself. And if we want to be more complicated, we can make expressions, like:
What is the value of this expression?
3+2
(This isn't a trick question.)
Answer: 5.
Definition of "Variable"
So what is a variable? One simple definition is:
A variable is something which could have a value, but we haven't decided yet what the value is going to be.
In elementary algebra, the as-yet undetermined values are almost always numbers.
Because we haven't yet decided what the value of a variable is, we can't write down the value, so we have to write the variable some other way, and the usual way to do this is to give the variable a name, and the most common name for a variable is the letter "x", which is usually written in italics, like so: x
Here is an example of an expression containing the variable x:
x + 3
The Values of Expressions Containing Variables
So what is the value of x?
The basic answer is that it is 3 more than whatever is. But since we haven't yet decided what the value of is yet, we can't know what the value of is.
What's the Point of Talking About Unknown Values?
What is the point of taking some number that we haven't decided what it is, and then adding 3 to it, to get another number that we don't know what it is, but which must be 3 more than the number we started with?
There are actually a few different reasons for wanting to do this. The most common reasons are among the following:
The number has a value, but we don't yet know what it is. We have to talk about the unknown value in order to find out what it is, and in order to talk about it, we have to give it a name, like, for example, "x".
The number x has a value, but we just don't care what it is.
We want to say something about a number x that is true for all numbers.
Having More Than One Variable
Give a mathematician a hard problem, and sometimes they'll sit down and think up a harder problem, even before they have properly solved the first problem.
The way to do this with variables is to have more than one variable.
Each variable has to have a name, and different variables have to have different names. Following a common mathematical rule of thumb, which is:
If you have to use a different letter, then use the next available letter of the alphabet,
a second variable will generally be called "y", also written in italics, like so: y
Here's an example of an expression using two variables and :
x + y
What this expression means, is take the unknown value of x and add it to the unknown value of y, to get a third number, whose value is of course not known, but which is equal to the sum of the unknown values of x and y.
Study Unit 1.3 - Fractions
Algebraic fractions are simply fractions with algebraic expressions on the top and/or bottom.
When adding or subtracting algebraic fractions, the first thing to do is to put them onto a common denominator (by cross multiplying).
Solving equations
When solving equations containing algebraic fractions, first multiply both sides by a number/expression which removes the fractions.
Study Unit 1.4 - Powers and Roots
Powers are used when we want to multiply a number by itself repeatedly.
Powers
When we wish to multiply a number by itself we use powers, or indices as they are also called.
For example, the quantity 7×7×7×7 is usually written as 74. The number 4 tells us the number
of sevens to be multiplied together. In this example, the power, or index, is 4. The number 7 is
called the base.
Example
62 = 6× 6 = 36. We say that ‘6 squared is 36’, or ‘6 to the power 2 is 36’.
25 = 2× 2 × 2 × 2 × 2 = 32. We say that ‘2 to the power 5 is 32’.
Your calculator will be pre-programmed to evaluate powers. Most calculators have a button marked
xy, or alternatively ˆ . Ensure that you are using your calculator correctly by verifying that
311 = 177147.
Square roots
When 5 is squared we obtain 25. That is 52 = 25.
The reverse of this process is called finding a square root. The square root of 25 is 5. This is
written as √2 25 = 5, or simply √25 = 5.
Note also that when −5 is squared we again obtain 25, that is (−5)2 = 25. This means that 25 has
another square root, −5.
In general, a square root of a number is a number which when squared gives the original number.
There are always two square roots of any positive number, one positive and one negative. However,
negative numbers do not possess any square roots.
Most calculators have a square root button, probably marked √. Check that you can use your
calculator correctly by verifyingthat √79 = 8.8882, to four decimal places. Your calculator will
only give the positive square root but you should be aware that the second, negative square root is
−8.8882.
An important result is that the square root of a product of two numbers is equal to the product of
the square roots of the two numbers. For example
More generally, √ab = √a × √b
However your attention is drawn to a common error. It is not true that √a + b = √a + √b.
Substitute some simple values for yourself to see that this cannot be right.
Exercises
1. Without using a calculator write down the value of √9 × 36.
2. Find the square of the following: a) √2, b) √12.
3. Show that the square of 5√2 is 50.
Answers
1. 18, (and also −18). 2. a) 2, b) 12. 3. (5√2 )2 = 52 × (√2 )2 = 25 × 2 = 50
Cube roots and higher roots
The cube root of a number, is the number which when cubed gives the original number. For
example, because 43 = 64 we know that the cube root of 64 is 4, written √3 64 = 4. All numbers,
both positive and negative, possess a single cube root.
Higher roots are defined in a similar way: because 25 = 32, the fifth root of 32 is 2, written √5 32 = 2
Exercises
1. Without using a calculator find a) √3 27, b) √3 125.
Answers
1. a) 3, b) 5.
Study unit 1.5 - Ratios, proportions and percentages
We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to" -- we say "the ratio of something to something else" --
