It’s the Law: Properties of Real Numbers
The Order of Operations
Solve the following problem:
Solution 1:
Solution 2:
3 x 8 = 24
3 x 8 = 24
3 x 8 – 20 ÷ 10 + 5
24 – 20 = 2 2 ÷ 10 = .2
20 ÷ 10 = 2 24 – 2 = 22
.2 + 5 = 5.2
22 + 5 = 27
Two very different answers…Which one is correct?
By Mathematical Law we must follow the Order
of Operations .
Parentheses – Do any problem that appears within parentheses
Ex: (2 + 3) + 4 x 5 =
(6) + 4 x 5 =
Expon ents – Solve any exponents
Ex: 5 – 2 + 42 =
5 – 2 + 16 =
Multiply and Divide – Do all multiplication and/or division problems from left to right
Ex: 3 x 4 ÷ 2 =
Ex: 4 ÷ 2 x 3 =
3 x 4 = 12
4÷2=2
12 ÷ 2 = 6
2x3=6
Add and Subtr act – Do all addition and subtraction problems from left to right
Ex: 2 + 5 – 3 =
Ex: 2 – 5 + 3 =
2+5=7
2 – 5 = -3
7–3=4
-3 + 3 = 0
To remember the order people then came up with “Please Excuse My Dear Aunt Sally”
Back to our original problem.
3 x 8 – 2 ÷ 10 + 5
By order of operations we need to multiply then divide
3 x 8 – 20 ÷ 10 + 5
24 –
2
+5
Then we need to subtract then add
22 + 5 = 27
Solution #2 was correct
Zero Property of Multiplication states that any number multiplied by zero is
equal to zero.
5x0=0
0x9=0
33 x 8 x 0 = 0
Identity Property of Addition or Multiplication states that any number added
to zero or multiplied by 1 is equal to the original number. It’s like looking into a
mirror, same image comes back through.
4+0=4
0 + 72 = 7
23 x 1 = 23
1x9=9
Inverse Property of Multiplication states that any number
multiplied by its inverse is equal to 1.
1
3 4
7! =1
! =1
7
4 3
Commutative Property of Addition or Multiplication states that
no matter what order the addends or factors appear they both
will equal the same sum or product.
John is talking to Suzy. Suzy is talking to John. They are both
talking to one another.
5 + 7 = 7 + 5 = 12
4 x 9 = 9 x 4 = 36
Associative Property of Multiplication or Addition states that no matter
what the grouping of the addends or products they both will equal the same
sum or product.
(4 + 3) + 7 = 4 + (3 + 7) = 14
2 x (6 x 3) = (2 x 6) x 3 = 36
Distributive Property states that multiplying a sum by a number gives the same
result as multiplying each addend by the number and then adding the products
together. The outside number gets distributed to each inside number.
5 x (3 + 4) = 5 x 3 + 5 x 4 = 35
2 x (5 + 9) = 2 x 5 + 2 x 9 = 28