A social law is a rule that forbids or requires a given conduct of all members of a community. Laws are established by custom or through adoption by a legislature, that is, a group of people given the power to make laws. A mathematical law is different from a social law.
The algebraic laws are rules which, if followed, will assure the students that the procedures being used in a calcululation will result in a correct solution. A mathematical law, then is a principle or rule that, if applied throughout a body of mathematics, or to all members of a set, the mathematical operation performed is guaranteed to result in a correct solution
In other words, a mathematical law is a procedure, or rule, that if followed, will guarantee that the math involved will be performed correctly. Not performing mathematics according to a law will result in an incorrect solution.
It may surprise some students that it is possible to perform mathematical operations that result in incorrect answers. The laws of mathematics were developed by professional and highly expert mathematicians who staked their careers on the fact that, if followed conscientiously, their mathematical procedures would result in arithmetic and algebraic solutions that were correct in every way. Sometimes, a mathematical law can be looked upon as a mathematical operation, or procedure, that you can follow in a math problem and be guaranteed your math will come out correctly,
The question that may come up when considering mathematical laws is what kind of procedures can you do that will result in your mathematical calculations turning out wrong? Well, there are some things that you have to know about numbers in order to find out how you can perform a wrong mathematical operation. The first thing that you have to realize is that in order for you math to work the numbers that you are working with have to be linear. This means that each number that you are involved with must have the same weight (or value from it's beginning to it's end) as the other numbers you are calculating with. This refers to the fact that, if you pile up 16 1 lb. bags of salt on top of each other, each bag should have exactly 1 lb. of salt in it. You may say "of course, we know that." But, this is correct in arithmetic and algebra. However, there are branches of mathematics in which the weight of the numbers changes with increasing value. Logarithms and sine-curves are two branches of mathematics in which the successive numbers progressed change their relative weights as they increase. These kind of number intervals are called nonlinear. If you try to add these numbers on the assumption that they are algebraic (linear) numbers you will fall into error with your mathematical calculations and have all your work turn out wrong. This is a disaster when you are expecting that you math, that is supposed to be so correct, is actually wrong because you are doing something that is not proper for this kind of math you are using. The reason you would be performing an error is because you would not be following a pertinent mathematical law. In the sections that follow you will be acquainted with the commutative, associative, distributive properties, or laws, of mathematics.
The Commutative Law
The commutative law of algebra is, in concept, related to the word commute. In order to understand the commutative law let us consider an example of the word commute, which refers to regular travel, usually over a considerable distance, to and from work.
Doris lives in Harrison NJ and worked in New York City. The distance from her house to work was about 7 miles, as the crow flies, but, took considerable time to get there by car due to traffic delays. The PATH trains regularly run from Harrison to NYC, so Doris decided to take the train to and from work. On the average, the trip usually takes about 35 minutes each way. Ideally, the trip should take the same time going to work (35 minutes) as it takes coming from work (35 minutes). However, because of the heavy volume of riders at quitting time, Doris found that the return trip usually took 12 to 15 minutes longer than the trip to work.
The important point here is that the track distance traveled, each way, is the same. But, because the train is delayed at stations (picking up passengers) the same mileage traveled takes longer to traverse, or cover. Let us see what this means in terms of the algebraic concept of the commutative law.
As stated previously, the ideal concept of commutation (commuting) would involve the same time of travel to work as from work. Let us assume that this were the case: Assume that Doris traveled 35 minutes to work and 35 minutes from work, back to Harrison. Mathematically, this would signify a factual commutation
A factual, genuine, or exact mathematical commutation occurs when the value of the number under consideration traversed, or spanned, in one direction, is the same as the number spanned in the return direction. When the distance traveled in one direction is the same as the distance traveled over the return direction the mathematics used to represent the distances is said to commute.
Any mathematical calculations involving the distance Doris's train traveled from Harrison to NYC and from NYC to Harrison would commute because the distance is the same each way. However, any mathematical calculations involving the time taken to cover the trip to NYC and from NYC would not commute because the time to go to NYC is different from the time to return from NYC. Mathematical calculations involving the time for the trip back and forth would not commute. These calculations would be said to be anti-commutative.
The commutative law is expressed mathematically as
ab = ba
This is read as "a times b equals b times a." As described above, this means that multiplying two numbers in a forward direction results in the same value as multiplying the numbers in the reverse direction. In arithmetic and algebra this process holds because the values of the numbers are the same when going forward, or up, as they are when they go backward, or down. Numbers don't change value, or weight (a 5 has the same weight as a 9, etc.), so multiplication results in the same product whether you are multiplying forward or multiplying in reverse.
The same holds for addition. The commutative law of addition is
a + b = b + a
Here the commutative law states that two numbers added in one direction result in the same value as when added in reverse. This is true as long as there is no factor involved that causes a change when the numbers are added in reverse. As stated above, if the time of the trip to work is different than the time of the trip to return from work, the addition of the times will not be commutative. But, if the values of a and b are distances, the commutative law will hold because the distance to work and the distance from work cannot change over the same railroad track.
The central concept of the commutative law may be succinctly stated as:
Whenever two variables have the same values in the forward as they do in the reverse direction the variables commute. When two variables have different values in the forward direction than they do in the reverse direction t he variables are said to be anti-commutative.
PROBLEM 1: Apply the commutative properties to
(b) (-6) + 8
(a) (7)(-4) = (-4) (7)
-28 = - 28 (Same answer)
(b) (-6) + 8 = 8 + (-6)
2 = 2 (Same Answer)
Because the values are the same when the order of the numbers is reversed the multiplied numbers in (a) and the added numbers in (b) commute.
Subtraction And Division Are Not Commutative
PROBLEM 2: Show that subtraction and division are not commutative by applying the commutative law to
(a) (6) - (9)
(b) 21 ÷ 7
(a) (6) - (9) = -3, (9) - 6 = 3
Different answers (-3 3) In words, minus three is not equal to plus 3
(b) 21/7 = 3, 7/21 = 1/3
Different answers 3 1/3
The Associative Law
The associative law states that changing the grouping of numbers in a multiplication or addition will not change the value of the answer obtained. The Associative law is written as
+ b + c = (a + b) + c = a + (b + c)
PROBLEM 3: Explain the associative properties connected with
Changing the grouping of the terms in an addition does not change the value obtained for the addition. Adding 6 + (4 + 9) gives the same answer as adding (6 + 4) + 9. In an arithmetic or algebraic addition the order in which the numbers or values are added does not matter.
Changing the grouping of the terms in a multiplication does not change the value obtained for the solution. Multiplying 6 and 7 first and, then, multiplying that result with 9 gives the same answer as multiplying 7 and 9 first and, then, multiplying that result with 6. In an arithmetic or algebraic multiplication the order in which the numbers or values are multiplied does not matter. The principal reason that the associative law is taught in algebra courses is to show the students that he/she can rearrange the terms of an addition, or the factors in a multiplication, in any manner desired without causing an error in the mathematics. In short, the associative law gives you a principle that you can follow that tells you what arrangements, or moves, that you can make when you are working with complicated algebraic expressions.
Subtraction And Division Are Not Associative
PROBLEM 4: In the following problems show that the arithmetic and algebraic processes of subtraction and division do not follow the associative laws.
(a) ( 8 - 4) - 6
(b) [(-8) ÷ (2)] ÷ 4]
(a) (8 - 4) - 6 = 4 - 6
8 - (4 - 6) = 8 - (-2)
= 8 + 2
Since -2 10 (read as -2 is not equal to 10),
(8 - 4) - 6 8 - (4 - 6)
indicating that the associative law does not hold for subtraction
(b) Perform the operation within the parentheses first, results in,
[(-8) ÷ (2)] ÷ 4 = -8/2 ÷ 4
= -4 ÷ 4
Next, move the parenthesis to encompass the last two numbers. Then, performing the operation within the parenthesis, first, results in
(-8) ÷ [(2) ÷ (4)] = -8 ÷ 1/2
= -8 @ 2/1
= - 16
Since -1 -16,
[(-8) ÷ (2)] ÷ 4 (-8) ÷ [(2) ÷ (4)]
indicating that the associative law does not hold for division.
THE DISTRIBUTIVE LAW
The distributive law of arithmetic and algebra is:
a(b + c) = ab + ac
The distributive law states that when an enclosed expression like (b + c) is multiplied by a factor like a, the product can be obtained by multiplying each term of the expression by the factor and affixing the sign within the enclosed expression (in this case a +) between the products obtained.
PROBLEM 5: Show that the distributive law holds for the following:
5(3 + 6) = (5 @ 3) + (5 @ 6)
5(3 + 6) = 5(9) = 45
(5 @ 3) + (5 @ 6) = 15 + 30 = 45
Since both sides of the problem are equal, the distributive law holds for this equation.
The distributive law often permits a complicated expression on it's right side to be simplified into a simpler expression on it's left side. Sometimes the distributive law can be used to alter, or, change an expression on it's left side be altered into a different expression (a mathematical space). For example, through the distributive law,
(2x - 4y) = 7(2x) - 7(4y)
= 14x - 28y
converts 7(2x - 4y) into another expression which takes on the form 14x - 28y.
PROBLEM: Use the distributive law to simplify the following algebraic expression: