Fundamentals of Mathematics I

1 Fundamentals of Mathematics IKent State Department of Mathematical SciencesFall 2008 Available at: 4, 2008 Contents1 Real Numbers.. Exercises .. Addition.. Exercises .. Subtraction.. Exercises .. Multiplication.. Exercises .. Division.. Exercise .. Exponents.. Exercises .. Order of Operations.. Exercises .. Primes, Divisibility, Least Common Denominator, Greatest Common Factor.. Exercises .. Fractions and Percents.. Exercises .. Introduction to Radicals.. Exercises .. Properties of Real Numbers.. Exercises ..572 Basic Combining Like Terms.. Exercises .. Introduction to Solving Equations.

2 Exercises .. Introduction to Problem Solving.. Exercises .. Computation with Formulas.. Exercises ..763 Solutions to Exercises771 Chapter Real NumbersAs in all subjects, it is important in Mathematics that when a word is used, an exact meaning needs to be properlyunderstood. This is where we will you were young an important skill was to be able to count your candy to make sure your sibling did not cheat youout of your share. These numbers can be listed:{1,2,3,4, ..}. They are calledcounting numbersorpositive you ran out of candy you needed another number 0. This set of numbers can be listed{0,1,2,3, ..}. They arecalledwhole numbersornon- negative integers.

3 Note that we have usedset notation for our list. A set is just acollection of things. Each thing in the collection is called an element or member the set. When we describea set by listing its elements, we enclose the list in curly braces, {} . In notation{1,2,3, ..}, the ellipsis, .. ,means that the list goes on forever in the same pattern. So for example, we say that the number23is anelement of the set of positive integers because it will occur on the list eventually. Using the language of sets,we say that0is an element of the non- negative integers but0is not an element of the positive integers. Wealso say that the set of non- negative integers contains the set of positive you grew older, you learned the importance of numbers in measurements.

4 Most people check the temperature beforethey leave their home for the day. In the summer we oftenestimateto the nearest positive integer (choose the closestcounting number). But in the winter we need numbers that represent when the temperature goes below zero. We canestimate the temperature to numbers in the set{.., 3, 2, 1,0,1,2,3, ..}. These numbers are numbersare all of the numbers that can be represented on a number line. This includes the integers labeledon the number line below. (Note that the number line does not stop at -7 and 7 but continues on in both directions asrepresented by arrows on the ends.)Toplota number on the number line place a solid circle or dot on the number line in the appropriate : Sets of Numbers & Number LineExample 1 Plot on the number line the integer :Practice 2 Plot on the number line the integer :Click here to check your 3Of which set(s) is0an element: integers, non- negative integers or positive integers?

5 Solution: Since 0 is in the listings{0,1,2,3, ..}and{.., 2, 1,0,1,2, ..}but not in{1,2,3, ..}, it is an element of theintegers and the non- negative 4Of which set(s) is5an element: integers, non- negative integers or positive integers?Solution:Click here to check your it comes to sharing a pie or a candy bar we need numbers which represent a half, a third, or any partial amountthat we need. Afractionis an integer divided by a nonzero integer. Any number that can be written as a fraction is calledarational number. For example, 3 is a rational number since 3 = 3 1 =31. All integers are rational numbers. Noticethat a fraction is nothing more than a representation of a division problem.

6 We will explore how to convert a decimal to afraction and vice versa in section the fraction12. One-half of the burgandy rectangle below is the gray portion in the next picture. It representshalf of the burgandy rectangle. That is, 1 out of 2 pieces. Notice that the portions must be of equal numbers are real numbers which can be written as a fraction and therefore can be plotted on a number line. Butthere are other real numbers which cannot be rewritten as a fraction. In order to consider this, we will discuss decimals. Ournumber system is based on 10. You can understand this when you are dealing with the counting numbers. For example, 10ones equals 1 ten, 10 tens equals 1 one-hundred and so on.

7 When we consider a decimal, it is also based on 10. Consider thenumber line below where the red lines are the tenths, that is, the number line split up into ten equal size pieces between 0and 1. The purple lines represent the hundredths; the segment from 0 to 1 on the number line is split up into one-hundredequal size pieces between 0 and in natural numbers these decimal places have place values. The first place to the right of the decimal is the tenthsthen the hundredths. Below are the place values to the : ones: . : tenths: hundredths: thousandths: ten-thousandths: hundred-thousandths: millionthsThe number can be read thirteen and four hundred fifty-three thousandths.

8 Notice that after the decimalyou read the number normally adding the ending place value after you state the number. (This can be read informally as thirteen point four five three.) Also, the decimal is indicated with the word and . The decimal would be one andthirty-four ten-thousandths .Real numbers that are not rational numbers are calledirrational numbers. Decimals that do not terminate (end) orrepeat representirrational numbers. The set of all rational numbers together with the set of irrational numbers is calledthe set ofreal numbers. The diagram below shows the relationship between the sets of numbers discussed so far. Someexamples of irrational numbers are 2, , 6 (radicals will be discussed further inSection ).

9 There are infinitely manyirrational numbers. The diagram below shows the terminology of the real numbers and their relationship to each the sets in the diagram are real numbers. The colors indicate the separation between rational (shades of green) andirrational numbers (blue). All sets that are integers are in inside the oval labeled integers, while the whole numbers containthe counting : Decimals on the Number LineExample 5a) Plot on the number line with a black ) Plot with a green : For we split the segment from 0 to 1 on the number line into ten equal pieces between 0 and 1 and then countover 2 since the digit 2 is located in the tenths place.

10 For we split the number line into one-hundred equal pieces between0 and 1 and then count over 43 places since the digit 43 is located in the hundredths place. Alternatively, we can split upthe number line into ten equal pieces between 0 and 1 then count over the four tenths. After this split the number line upinto ten equal pieces between and and count over 3 places for the 3 6a) Plot on the number line with a black ) Plot with a green :Click here to check your 7a) Plot on the number line with a black ) Plot with a green : a) Using the first method described for , we split the number line between the integers 3 and 4 into one hundredequal pieces and then count over 16 since the digit 16 is located in the hundredths ) Using the second method described for , we split the number line into ten equal pieces between 1 and 2 andthen count over 6 places since the digit 6 is located in the tenths place.