Unit 1: Number Properties
In this unit, we will discuss some of the basic algebraic properties you may already know, that many math instructors call "common sense" properties. We see uses of these properties every day.
For example, the commutative property tells us we can rearrange the order of the numbers and still get the same result: 3 + 2 = 5, and 2 + 3 = 5. The same is true for multiplication: 2 × 3 = 6, and 3 × 2 = 6.
Other algebraic properties are less intuitive. For example, what happens when we multiply or divide a number by zero? Understanding these number properties will give you the foundation you need for this course.
Completing this unit should take you approximately 4 hours.
1.1: Commutative Law of Addition and Multiplication
It is important to note that the commutative property only holds for addition and multiplication. It does not work for subtraction or division. We can make a simple example to show this: 10 − 2 = 8, but 2 − 10 = −8.
The results are not the same when we change the order of the numbers in subtraction. Likewise, 20/2 = 10, but 2/20 = 0.10. Again, the results are not the same when we change the order of the numbers in division.
1.2: Associative Law of Addition and Multiplication
The associative law of addition and multiplication tells us that the grouping of numbers in addition and multiplication does not change the result. In math, when we put parentheses around a set of numbers, we do the calculation in the parentheses first. For example, 5 + (2 + 1) = 8. When we calculate this, we first calculate 2 + 1 = 3. Then, we add 5 to get 8. We can also write (5 + 2) + 1 = 8. Here, we first calculate 5 + 2 = 7, and then add one to get a sum of eight. We get the same result regardless of how we group the numbers.
The same law works for multiplication. We can compute (2 × 2) × 3 = 12. We can change the grouping and write 2 × (2 × 3) = 12. The placement of the grouping does not change the answer.
1.3: Identity Property of Addition
In math, the identity is a number that can be added or multiplied to another number to give the same number. The identity property of addition states that if you add zero to a number, the sum will be that number. That is, 0 + 5 = 5.
1.4: Inverse Property of Addition
The inverse property of addition states that a number plus its negative (or inverse) will equal zero. That is, 5 + −5 = 0.
1.5: Identity Property of Multiplication
Much like the identity property of addition (section 1.3), the identity property of multiplication states the identity for multiplication. For multiplication, the identity property states that if you multiply any number by 1, the answer is that number. For example, 3 × 1 = 3.
1.6: Inverse Property of Multiplication
The inverse property of multiplication tells us that any number times its inverse (1 divided by the number) will equal one. For example, 2 × ½ = 1.
1.7: Multiplication by Zero
When we multiply any number by zero, the result is always zero. For example, 15 × 0 = 0. Likewise, because of the commutative property, 0 × 15 = 0.
1.8: Dividing by Zero Is Undefined
Because division does not follow the commutative property, we must consider two cases when doing division with zero. First, we can divide zero by a number. Zero divided by any number is zero. For example, 0/5 = 0.
Now we must consider the case when a number is divided by zero. When we divide a number by zero, the answer is undefined. For example, 5/0 = undefined. In math, undefined means that there is no possible answer.
1.9: Distributive Property
The distributive property allows us to distribute a multiplier to a sum in a parentheses. For example, if we have 2 (3 + 5), we can rewrite this using the distributive property as: (2 × 3) + (2 × 5). This often makes it easier for us to do mental math, or to at least simplify a large calculation.
Unit 1 Assessment