The Algebraic Laws |

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Volume 1 Issue 2 May 1996 |

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

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

The commutative law of algebra is, in concept, related to the word

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

Any mathematical calculations involving the distance Doris's train traveled from Harrison to NYC and from NYC to Harrison would

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

The central concept of the commutative law may be succinctly stated as:

(a) (7)(-4)

(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

(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 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)

(a) 6 + (4 + 9) = (6 + 4) + 9

(b) [(6)(7)] (8) = (6) [(7)(8)]

(a) 6 + (4 + 9) = 6 + 13

= 19

(6 + 4) + 9 = 10 + 9

= 19

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.

(b) [(6)(7)] (8) = 42 @ 8 = 336

(6) @ [(7)(8)] = 6 @ 56 = 336

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

(a) ( 8 - 4) - 6

(b) [(-8) ÷ (2)] ÷ 4]

(a) (8 - 4) - 6 = 4 - 6

= -2

8 - (4 - 6) = 8 - (-2)

= 8 + 2

= 10

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

= -1

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 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.

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

(2x - 4y) = 7(2x) - 7(4y)

= 14x - 28y

converts 7(2x - 4y) into another expression which takes on the form 14x - 28y.

(-4x2 + xy - 4y3)(-6xy)

(-4x2 + xy - 4y3)(-6xy)

= (4x2) (-6xy) + (xy)(-6xy)-( 4y3)(-6xy)

= -24x3y - 6x2y2 + 24xy4

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