Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
Algebraic Laws and Geometric Postulates
Algebraic laws are laws that tell us how things add, subtract, multiply, divide, and otherwise combine together. Geometric postulates are those basic truths that are the basis for other theorems. It is important to learn and understand these laws and postulates, because once you know them, you can easily manipulate equations and solve geometric and algebraic formulas. In this video, we are going to cover the commutative, associative, distributive, reflexive, symmetric, and transitive laws. Keep watching to learn what they are and how to use them!
The commutative law tells us that we can add and multiply numbers in whatever order we like. Written algebraically, the commutative law says that x + y = y + x and x * y = y * x. For example, 1 + 2 is also equal to 2 + 1. The same thing goes for 1 * 2. 1 * 2 is the same as 2 * 1. The order does not matter. But remember, this only works for all addition or all multiplication. Once you mix them, you have to evaluate the expression following the order of operations.
While the commutative law tells us that we can add and multiply two numbers in any order, the associate law tells us that we can add and multiply three numbers in any order. Algebraically, it is written as x + (y + z) = (x + y) + z and x(yz) = (xy)z. Using an example, 1 + (2 + 3) is the same as (1 + 2) + 3 and 1(2 * 3) is the same as (1 * 2)3.
Basically, what this is saying is that we don't need to use the parentheses to force the order of our addition or multiplication if we have all addition or all multiplication. Evaluating our example, we see that both sides are equal. 1 + (2 + 3) = 1 + 5, which equals 6. (1 + 2) + 3 = 3 + 3, which equals 6 as well. The multiplication is equal as well. 1(2 * 3) = 1 * 6, which equals 6. (1 * 2)3 = 2 * 3, which also equals 6.
If we mix multiplication with addition along with a pair of parentheses like x(y + z), then the distributive law applies and tells us that x distributes to the y and the z. Algebraically, x(y + z) becomes xy + xz. As an example, the 2 in 2(3 + 4) distributes to the 3 and the 4 to become 2 * 3 + 2 * 4.
The reflexive law is rather obvious as it tells us that a number is equal to itself. Using variables, x = x. Using numbers, 1 = 1.
Similar to the reflexive law, the symmetric law tells us that if one variable equals another, then the other variable equals the first. Using variables, if x = y, then y also equals x. Using numbers, if 1 = 1, then 1 also equals 1. Using both numbers and variables, if 3 = b, then b also equals 3.
The transitive law tells us if one item equals a second item and the second item equals a third, then the first item also equals the third item. Algebraically, we have if x = y and y = z, then x = z. Using both numbers and variables, we have if x = y and y = 3, then x = 3.
What have we learned? We've learned that algebraic laws are laws that tell us how things add, subtract, multiply, divide, and otherwise combine together, and geometric postulates are those basic truths that are the basis for other theorems:
- The commutative law tells us x + y = y + x and x * y = y * x.
- The associative law tells us x + (y + z) = (x + y) + z and x(yz) = (xy)z.
- The distributive law is written as x(y + z) = xy + xz.
- The reflexive law tells us that any number is equal to itself: x = x.
- The symmetric law says if x = y, then y also equals x.
- Last, but not least, the transitive law tells us if x = y and y = z, then x = z.
After you've reviewed this video lesson, you should be able to:
- Define algebraic laws and geometric postulates and explain why it is important to understand them
- Describe the commutative, associative, distributive, reflexive, symmetric and transitive laws
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