Real Numbers and Algebraic Expressions

Transcription

1 College Algebra - MAT 161 Page: 1 Coyright 2009 Killoran Real Numbers and Algebraic Exressions Idea #1 Infinity- a never ending rogression Is there an infinity of sand grains on the Outer Banks of North Carolina, or could it be counted? Think a bit on this second one, could this be done, could you count every grain of sand?? The answer is YES, because eventually (in a few million years) every grain of sand could be counted! Thus there is not an infinity of sand on the beach, even though you ersonally might not be able to count them all. An infinity assumes that once you think that you reached it, there is always one more, and one more, and... Idea #2 The Real Numbers-the basis of most of mathematics is Numbers and Numeration. Natural Numbers (counting numbers) These were used by the oldest of eole and was easy to understand. To list them all would be imossible so we use something called Set Notation to accomlish this. Using ellises to indicate a continuing attern to infinity the Natural numbers would be: N D f1; 2; ; 4; 5; :::g Whole Numbers Here is a huge addition to the number system, the concet of ZERO. We will find that for thousands of years the concet of zero was avoided. Think how you would answer the question and you will have an idea of why zero was a comlex creation: How many oranges are in an emty bowl? Why NONE of course. W D f0; 1; 2; ; 4; :::g Integers (the essence of the idea of Negative) The idea of Negative, the absence of "stuff" is necessary to talk about simle Accounting used in a Check Book. Deosits are. / and Checks written are. / to the account. I D f:::; ; 2; 1; 0; 1; 2; ; :::g Rational Numbers better known as fractions. / where &q are integers. They can also be reresentative of decimal numbers that terminate. / or that reeat. / : There is no list that can be comiled, like above with the integers. So they are just designated as any number q in above sentence. So what kind of numbers exist. The Greeks (Pythagorean s) thought that all numbers could be reresented as a rational number (length for their math). They found though with use of the Pythagorean Theorem (the square of the hyotenuse equals the sum of the squares

2 College Algebra - MAT 161 Page: 2 Coyright 2009 Killoran of the legs on any right triangle, c 2 D a 2 C b 2 / they found a number (length) that could not be reresented by a rational number, mainly 2. This tye of number is called: Irrational Numbers these are numbers that cannot be reresented as an integer. Thus they are decimal numbers that do not terminate and do not reeat. One of the most famous of these irrational number is Pi D : ::: Think: If some game show asks you to tell them any Real Number, what would you tell them? Most eole ick a Counting Number, very few will say " 1": Why do you think this is so??? Otional- How do we know that irrational numbers are not really very comlex rational numbers? How do we know that somewhere down the long line of decimal laces that these numbers don t start reeating ( D : 141 6::::::/?? Well here is an examle of Deductive Reasoning Backwards (roof by contraositive). We will make an assumtion that is the oosite of what we are trying to rove and show this leads to a contradiction of what we know to be true. Then what must be true is what we were trying to show in the first lace. Prove: Irrational numbers are not Rational Numbers Assume that 2 D q ; where &q are integers and the fraction q common factors! is in lowest terms, no Thus 2 D q squaring both sides: 2 2 D q 2 2 D 2 q 2 2q 2 D 2 Any number that has a factor of 2 in it is called EVEN. Thus 2 is even and only even squares are even thus is even. D 2k/. 2q 2 D.2k/ 2 2q 2 D 4k 2 q 2 D 2k 2 Thus q 2 is even and by above statements so is q.q D 2m/ so:

3 College Algebra - MAT 161 Page: Coyright 2009 Killoran q D 2k 2m D k m But this is a contradiction of our original statement that was in lowest terms. Thus what q we assumed to be true is FALSE and its oosite must be true!! 2 is not a rational number!!!! 1 Absolute Value Proerties: You should know that j 7j D 7 and jj D but what about j 7j We cannot do the following: j 7j 6D C 7 That is not the what an absolute value does to a number. You have to remember that the exression. 7/ is actually just one number, and under the absolute value, this number must come out as a ositive value. Thus one way of figuring this out would be to use a trick (that we will use often in this class. What is the difference between the answer for. 7/ and.7 /? Well the first one,. 7 D 4/ and the second one.7 D 4/. That is they are only different by a sign! Rule: a b D.b a/ Thus to figure out j 7j we only need to make sure the Larger one comes first in the subtraction (thus guaranteeing that the answer will be ositive!) j 7j D 7 D 4 Evaluate the following absolute value 2 We only need to find out which of the two values above is Larger, and make sure we write it in the first sot. We know that the 1 D 1 and 4 D 2 so 1 < < 4 1 < < 2 so we know for sure that < 2 thus their difference 2 is negative To make it ositive we just change the order of the subtraction so... 2 < 0 : 2 D 2

4 College Algebra - MAT 161 Page: 4 Coyright 2009 Killoran Examle 1 Evaluate: j2 6j Here we need to know which is larger, the 2 or the 6: We should know that :14 and we know for sure that > thus 2./ > 2./ and 2 > 6: So we will leave the order with the larger value first: j2 6j D Distance between two numbers on a number line How many units are there between A D and B D 5? Plot the numbers on a number line and count the number of saces that searate them. 8 units x Thus we found d.a; B/ D 8 Well if the numbers get large or if they are reresented by letters then that method will not work. So how else do we find the answer of 8 u above? How about j 5j or j5. /j? Both of these seem to work! So d.a; B/ D ja bj Evaluation of Formulas and Functions My method of substitution of values into formulas is much different than the way most students would do them. I do them this way because I know what is coming and if you get used to it this way then what is coming will not be hard at all. Stes for Substitution: Relace all known variables with. / (emty set of arentheses) Put values into roer. / Simlify using order of oerations (and any algebraic maniulation)

5 College Algebra - MAT 161 Page: 5 Coyright 2009 Killoran Examle 2 Find the Volume of a Cylinder given that it has a height.h/ of 12 inches and a radius.r/ of 5 inches. V D r 2 h V D. / 2. / V D.5/ 2.12/ V D 00 in 942: 5 in Examle Given that x D 2 and y D 4 find z given that z D 1 C 2y x C xy z D 1 C 2. /. / C. /. / z D 1 C 2.4/. 2/ C. 2/.4/ z D 1 C 8 C 24 8 z D 25 Examle 4 Given that a D 2; b D ; and c D 4 using x D b b 2 4ac 2a x D. / C. / 2 4. /. / 2. / x D. / C. / /.4/ 2. 2/ x D C 9 C 2 4 x D 9 C 2 4 x D 0:85 4 Proerties of Real Numbers Commutative: a C b D b C a and ab D ba Associative: a C.b C c/ D.a C b/ C c and a.bc/ D.ab/ c Distributive: a.b C c/ D ab C ac or x y C zy D y.x C z/ (better known as factoring) Identities: a C 0 D a and 1 a D a 1 Inverses: a C. a/ D 0 and a D 1 a A note about "Combining Like Terms".

6 College Algebra - MAT 161 Page: 6 Coyright 2009 Killoran There is no mathematical roerty called combine like terms. above does allow us to do this shortcut. But one of the roerties Think about how you add 7x C 5x 1) Your mind figures that they have a common term, called x 2) You romtly "ignore" the term and add the 7 and 5 together ) You ut back the common term 1/ 7x C 5x 2/.7 C 5/ / 12x This is just the Distributive Proerty in disguise: 7x C 5x.7 C 5/ x 12x **When students make the mistake of adding x C x and getting x 2 they are forgetting that there are 1 s in front of these x 0 s (by the identity roerty). So when in doubt, ut a 1: 1x C 1x D. / x D 2x 5 Simlifying Exressions /, grou common terms together (commutative and associa- Distribute first if there are. tive, and then simlify....2x 4/ C 4.x C 5/ 6x 12 C 4x C 20 6x C 4x 12 C 20.6 C 4/ x 12 C 20 10x C 8 2x.x 5/ C 7.2x C 4/ 10x 6x 2 C 14x C 28 6x 2 C 24x C x C 1/ C 4 8x 2 C 4 5 8x

7 College Algebra - MAT 161 Page: 7 Coyright 2009 Killoran 8 5 [ C 2.4x C 7/] 8 5 [ C 8x C 14] 8 5 [8x C 17] 8 40x 85 40x 77 6 A Few things about Notation 6.1 The Definition of Subtraction: a b D a C. b/ thus under the Commutative Proerty: a b D b C a 6.2 Multilying Fractions: a b c d D ac bd 6. Dividing Fractions: a b c d D a b d c thus a b D a 1 b D a 1 1 b D a b 6.4 Dealing with Fractions and Signs: a b D a b D a b The negative "value" is not changed by the osition of the negative "sign" (as long as one number has that sign and definitely not both, that would be a ositive value). Examle 5 6y C 9x 2 D. 6y C 9x 2/ D 6y 9x C 2

8 College Algebra - MAT 161 Page: 8 Coyright 2009 Killoran 6.5 Vertical Line Notation While looking at this last examle it would be rudent to notice that when we "moved" the negative sign to the Numerator that we enclosed the Numerator with arenthesis. This is because there is an old math notation (before arenthesis) where a vertical line was a grouing symbol. This shows u in fractions (like the above examle) as well as in Root Notation: x 4 where the line grous the x 4. In old notation.2x 1/.x C 4/ was written as 2x 1 x C 4. So when you see these Vertical Lines above or below an exression, there are truly a set of arenthesis around them, grouing them together. Examle 6 2x 4 D.2x 4/ Examle 7 2 x2 5x D 2 x2 5x