RUNNINGHEAD: SIMPLIFYING EXPRESSIONS
In arithmetic we use only positive numbers and zero, but with algebra, we use both positive and negative numbers. The numbers we use in algebra are called the “real numbers” or integers {… , -3, -2, -1, 0, 1, 2, 3…}. In this paper I am going to explain the properties of real numbers using three examples. I will also be explaining how to solve these examples step by step, all while discussing why these properties are so important to begin with. The properties of real numbers are the commutative, associative, identity, and additive inverse properties of addition, distributive law, and the commutative, associative, identity, and the multiplicative inverse (reciprocal) of multiplication. What these properties mean is that order and grouping don’t matter for addition and multiplication, but they certainly do matter for subtraction and division. In this way, addition and multiplication are much cleaner than subtraction or division. This is extremely important when talking about simplifying algebraic expressions. Often what we will want to do with an algebraic expression involves rearranging it somehow, and combining like terms.
If the operations are all addition and multiplication, we don’t have to worry so much that we might be changing the value of an expression by rearranging its terms or factors. Fortunately, we have the option to think of subtraction as an addition problem (adding the opposite), and we can always think of division as a multiplication problem (multiplying by the reciprocal).
The Essay on Basic Algebraic Properties of Real Numbers
... of addition and multiplication. Basic Algebraic Properties: Let and denotes real numbers. (1) The Commutative Properties (a) (b) The commutative properties ... associative properties, expressions such as and makes sense without parentheses. (3) The Distributive Properties (a) (b) The distributive properties ... then by definition It may be noted that Division by zero is not allowed. When is written in ...
The additive identity property of addition has the rule that you can add zero to any number and its original identity will always remain the same. The additive inverse property rules that any numbers plus that numbers negative will equal zero. ex. 5 + -5=0 or -5 + 5=0
You may have noticed that the commutative and associative properties read exactly the same way for addition and multiplication, as if there was no difference between them other than their notations. The property that makes them behave differently is the distributive property, because multiplication distributes over addition, and not vice-versa. The distributive property is extremely important, and it is impossible to even begin to understand algebra without being thoroughly familiar with it. The distributive property is used to apply multiplication across two or more terms that are surrounded by parenthesis. The final result of distribution is an equation that no longer has parenthesis. The multiplicative identity is 1. The rule for this is that any number multiplied by 1 will always retain its identity (as in + 0 for additive identity).
The inverse property of multiplication states that any number times its reciprocal will equal 1. ex. 8 * 1/8 = 8/1 * 1/8 = 1
These properties will be used to solve the following equations… A) 2a(a – 5) + 4(a – 5) The given expression
2a^2 – 10 + 4a – 20The distributive property removes the parentheses 2a^2 + 4a – 10Like terms are combined by adding coefficients
The problem is now completely simplified.
B) 2w – 3 + 3(w – 4) – 5(w – 6) The given expression 2w – 3 + 3w -12 – 5w – 30The distributive property removes the parenthesis 2w + 3w – 5w – 30 – 12 – 3Like terms are put together by rearranging using the commutative property 5w-5w -45Continue combining like terms
The Term Paper on Cultural Identity in The Namesake
The Namesake illustrates several elements of transition that are common to the stories of immigrant families and their children. As shown in the film, the first generation connects with their cultural identity and roots to a far greater degree and density than their children do. The second generation exists between two realities of culture including their ethnic heritage and the world they live in ...
= -45The problem is now completely simplified.
C) 0.05(0.3m + 35n) – 0.8(-0.09n – 22m) The given equation 0.015m + 1.75n + 0.072n + 17.6mThe distributive property removes parenthesis 17.615m + 1.822n Combine like terms
The problem is now completely simplified
In conclusion, we can see that real numbers and their properties are extremely important. Many times we use these skills and don’t even think twice because we have learned them throughout school as a child. You should always keep these properties in mind when solving expressions, because if the rules are not used, your answer will be incorrect. Knowing how numbers behave in relation to one another and their pre-determined notations is probably the most imperative skill in algebra. The rules learned from the properties of real numbers are vital to understanding all concepts in math and they are essential to ensuring accuracy.
