Number Properties, Order of Operations |
Properties of Real Numbers
The properties we discuss here are the strict rules of behavior for constants or Real numbers and the algebraic variables (letters) we use to represent Real numbers. They define 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.
1. The Commutative Property
The commutative property says that order doesn't matter when we multiply and/or add.
So 5 + 9 is exactly the same as 9 + 5. Also, 5 × 9 = 9 × 5.
To commute is to go back and forth -- and that's what this property tells us we can do.
2. 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.
If a, x and y are any three Real numbers, then: a(x + y) = ax + ay So, 3(5 + 7) = (3 × 5) + (3 × 7) = |
And since the commutative property says that order doesn't matter when we multiply and add, we can say:
(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 everything in 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 whenever possible. Another convention is to write expressions in ascending or descending powers of the variable.
So 3x² – 5x + 57 is more acceptable than – 5x + 57 + 3x² or – 5x + 3x² + 57.
We order the powers of x from highest down to lowest or from lowest to highest.
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 = 9.
Since we multiply by NEGATIVE 3, and then SUBTRACT,
we get POSITIVE 27 from the second operation.
For explanations on algebra in general and why 2 negatives make a positive --
study The Algebra Primer file in the Algebra MathRoom.
3. The Associative Property
This property says that we can add a list of values in any order or "grouping", as long as we "associate" or "group" them all. In other words (7 + 9) + 3 = 7 + (9 + 3) = (7 + 3) + 9
If a, b and c are any three Real numbers, a + (b + c) = (a + b) + c = (a + c) + b |
Now, when we combine both the distributive and associative properties, we can do the indicated 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 we learn the order in which to do the different operations.
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4. 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.
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
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.
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Practice
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 insert a pair of brackets to make it true:
a) 7.6 + 5.3 × 1.2 + 0.7 = 17.67 | b) 7.6 + 5.3 × 1.2 + 0.7 = 16.18 |
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) |
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Solutions
1. Simplify:
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 insert a pair of brackets to make it true:
a) 7.6 + 5.3 × (1.2 + 0.7) = 17.67 | b) ( 7.6 + 5.3) × 1.2 + 0.7 = 16.18 |
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 = 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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