| Number Properties, Order of Operations |
Properties of Real Numbers
The properties we discuss here are really the rules of behavior for Real numbers and the algebraic variables (letters) we use to represent Real numbers. They describe how the elements of algebra behave in relation to each other and to their position and order in the expressions and equations we simplify and solve.
reminder: In algebra, we indicate a × b as simply ab, or a ( b ).
The Commutative Property
The commutative property says that order doesn't matter when we multiply and/or add two numbers together. 15 + 3 always gives exactly the same results as 3 + 15. Also, we get the same product when we multiply 2 numbers from left to right as we do when we multiply them from right to left. So, 15 × 3 always gives exactly the same results as 3 × 15.
| The Commutative Property If a, and b are any two Real numbers, then: a + b = b + a and: a × b = b × a |
To commute means to go back and forth, and that's exactly what this property allows us to do.
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The Distributive Property
This property tells us how to "remove brackets" -- or "multiply out the brackets".
To distribute is to hand out -- to "spread" out -- and that's exactly what this property says.
| The Distributive Property If a, x and y are any three Real numbers, then: a(x + y) = ax + ay So, 3(5 + 7) = (3 × 5) + (3 × 7) = 15 + 21 = 36 |
And since the commutative property says that order doesn't matter when we multiply and add, we can reverse the order of the terms we have to multiply:
(x + y)a = ax + ay
( 3 + 8 )2 = (3 × 2) + (8 × 2) = 6 + 16 = 22
Notice how we distribute the multiplication by "a" and by "2" on the contents of the bracket that precedes or follows it. Notice also how we write the variables (letters) in alphabetical order.
Even though xa + ya is exactly the same as ax + ay,
we must respect mathematical convention so we write the variables in alphabetical order.
A common mistake for beginners is to ignore the sign of the multiplier.
For example: – 3(6 + 9 ) = (– 3 × 6 ) + (– 3 × 9 ) =
– 18 + ( – 27) = – 18 – 27 = – 45.
remember that + (– 27) is really + 1 × (– 27) -- we don't write the 1.
Since both numbers here are negative, they add to give negative 45.
Another convention to respect is to write the terms in ascending or
descending powers of the variable.
So 3x2 – 5x + 57 and 57 – 5x + 3x²
are more acceptable than
– 5x + 57 + 3x2 or – 5x + 3x² + 57.
We order the powers of x from highest to lowest or from lowest to highest.
For explanations on algebra in general and why 2 negatives make a positive study The Algebra Primer file in the Algebra MathRoom.
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The Associative Property
This property says that we can add a list of numbers in any order or "grouping", as long as we "associate" or "group" them all. In other words
| The Associative Property If a, b and c are any three Real numbers, |
This is the property that says we can get the same result when we add a list of numbers from the top down, or from the bottom up, or in whatever order we please, as long as we add all the numbers in the list.
Now, when we combine both the distributive and associative properties, we can perform operations on a bunch of numbers and then find the final result by adding or subtracting them in whatever order is best -- like this:
– 5(3 – 7) + 8 (6 + 3) = – 5(– 4) + 8 ( 9) = 20 + 72 = 82
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Exponents and Powers
Definition: To square a number means to multiply it by itself. 3² = 3 × 3 = 9
Definition: To cube a number means to multiply it by itself by itself. 3³ = 3 × 3 × 3 = 27
Note how the power tells us how many times to multiply the number by itself.
So 4 5 = 4 × 4 × 4 × 4 × 4
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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
Practice Exercise 1
State which property Commutative, Distributive or Associative is demonstrated by:
| a) 7 + 3 + 9 = 9 + 7 + 3 | b) (22 + 5) 9 = (22 × 9) + (5 × 9) | c) 90 + 14 = 14 + 90 |
| d) 971 × 4 = 4 × 971 | e) 25(6 + 7) = (25 × 6) + (25 × 7) | (solutions) |
Now we learn the order in which to do the different operations.
The Order of Operations
When we have to do a whole bunch of different operations on an expression or
in an equation, we must do so in a precisely defined order.
First, we do all calculations within Brackets
Second, we do all Exponentiation (raising to powers)
Third, we do all Multiplication and Division from left to right
And finally, we do all the Addition and Subtraction from left to right.
1. Simplify: 3.6 × (7 – 2) ÷ 3² + 1.3
1st: brackets ( 7 – 2 ) = 5.
We now have 3.6 × ( 5 ) ÷ 3² + 1.3
2nd: exponents 3² = 9.
Now we have 3.6 × ( 5 ) ÷ 9 + 1.3
3rd: multiply and divide 3.6 × ( 5 ) = 18 and 18 ÷ 9 = 2.
So now we have 2 + 1.3
Finally, collect "like" terms or add the numbers to get 3.3
2. Simplify: 24.6 – 3.1 × 7.4 + 11
Notice, here there are no brackets but we must do multiplication first, so:
Since 3.1 × 7.4 = 22.94, now we have
24.6 – 22.94 + 11 which we do from left to right to get 12.66.
3. Add brackets to make this statement true:
6 ÷ 3 + 3 × 3² = 45
We have to get 45 -- which is 5 × 9.
The last term is 3² which is 9 so we want the brackets to give us 5.
The solution is ( 6 ÷ 3 + 3 ) × 3² = 45 since we do brackets first
and 6 ÷ 3 + 3 = 2 + 3 or 5.
Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
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Practice Exercise 2
1. Simplify:
| a) 4² + 1.3 × 3 = | b) (5.3 – 4.7) × 0.2 = |
| c) 6.4 ÷ 1.6 × 0.1 = | d) 8.4 ÷ 2.1 – 0.7 + 6² = |
| e) ( 3.15 + 2.7 ) ÷ ( 5.2 – 1.4 ) = | f) 7.5 ÷ ( 2.5 – 1) – (5 – 3)² = |
2. Copy each expression then add brackets to make it true:
| a) 6 ÷ 3 + 3 × 3² = 9 | b) 6 ÷ 3 + 3 × 3² = 29 |
| c) 3.5 – 1.6 × 0.4 + 1.2 = 1.96 | d) 3.5 – 1.6 × 0.4 + 1.2 = 0.94 |
3. Simplify the expression on each side of the ? , then replace it with
< (less than), = (equal to), or > (greater than).
| a) 6² – 3² ? (6 – 3)² | b) 7² + 4² ? (7 + 4)² |
| c) 5 × (8 – 3) – 3 × 2 ? (15 – 4) × 6 ÷ 3 | d) (12 × 4) ÷ (8 + 8) ? (6 × 2) ÷ (3 + 1) |
Solutions
State which property Commutative, Distributive or Associative is demonstrated by:
| a) 7 + 3 + 9 = 9 + 7 + 3 Associative |
b) (22 + 5) 9 = (22 × 9) + (5 × 9) Distributive |
c) 90 + 14 = 14 + 90 Commutative |
| d) 971 × 4 = 4 × 971 Commutative |
e) 25(6 + 7) = (25 × 6) + (25 × 7) Distributive |
Practice Exercise 2
| a) 4² + 1.3 × 3 = 16 + 3.9 = 19.9 | b) (5.3 – 4.7) × 0.2 = 0.12 |
| c) 6.4 ÷ 1.6 × 0.1 = 0.4 | d) 8.4 ÷ 2.1 – 0.7 + 6² = 4 – 0.7 + 36 = 39.3 |
| e) ( 3.15 + 2.7 ) ÷ ( 5.2 – 1.4 ) = 1.53947 | f) 7.5 ÷ ( 2.5 – 1) – (5 – 3)² = 7.5 ÷ 1.5 – 4 = 1 |
2. Copy each expression then add brackets to make it true:
| a) 6 ÷ ( 3 + 3 ) × 3² = 9 | b) ( 6 ÷ 3 ) + ( 3 × 3² ) = 29 |
| c) (3.5 – 1.6) × 0.4 + 1.2 = 1.96 | d) 3.5 – 1.6 × (0.4 + 1.2) = 0.94 |
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3. Simplify the expression on each side of the ? , then replace it with
< (less than), = (equal to), or > (greater than).
| a) 6² – 3² ? (6 – 3)² = 36 – 9 ? (3)² becomes 27 > 9 |
b) 7² + 4² ? (7 + 4)² 49 + 16 ? (11)² becomes 65 < 121 |
| c) 5 × (8 – 3) – 3 × 2 ? (15 – 4) × 6 ÷ 3 5 × (5) – 6 ? (11) × 2 becomes 19 < 22 |
d) (12 × 4) ÷ (8 + 8) ? (6 × 2) ÷ (3 + 1) (48) ÷ (16) ? (12) ÷ (4) becomes 3 = 3 |
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