College Algebra FINAL EXAM Review, Fall 2010

College Mathematics FINAL EXAM ReviewPage 1 of 43

Section 1.1: Fractions

Section 1.1.1: Meaning of a Fraction

• A fraction is meant to represent the number of pieces of an object you take or use when you divide it into equal pieces. So, if you cut a pizza into eight equal-size pieces and then take five of them, we are talking about five of the eight equal-sized pieces. We write this amount of pizza as the fraction (of the pizza).
• There are three pieces to writing a fraction. Their names are:

• Fractions are usually called “rational numbers” in mathematical circles because a ratio is a name for when you compare how many pieces you have to a total number of pieces.

Section 1.1.2: Equivalent Fractions

• Fractions that have different numbers but that represent the same amount are called “equivalent fractions”.

• To convert one fraction to another equivalent fraction, you can either:

• Multiply both the numerator and denominator by the same number, or
• Divide both the numerator and denominator by the same number.

Reducing Fractions: “Reducing a fraction” involves writing the fraction as an equivalent fraction with smaller numbers in the numerator and denominator. Reducing a fraction until the numbers are as small as possible is called reducing the fraction to “lowest terms”.

• One way to reduce a fraction is just to do your best to see what you can divide top and bottom by.

• Another method for reducing fractions is to do a prime factorization of the numerator and denominator, then divide out (or “cancel”) all the common factors.

• Here is a list of the first twenty prime numbers to aid your factoring:

Equivalent fractions are also useful for comparing fractions when you don’t have a calculator.

To compare fractions:

• Write each fraction as an equivalent fraction, but with the same denominator.
• The bigger numerator belongs to the bigger fraction.

Section 1.1.3: Improper Fractions and Mixed Numbers

• To write a fractional amount that is between numbers bigger than one, we use either “improper fractions” or “mixed numbers”.

• An improper fraction is a fraction where the numerator is larger than the denominator. To see how to write pizzas as an improper fraction, you can think of the first pizza as being of a pizza (i.e., five of the five equal pieces), the second pizza as more, and then you also have the remaining of a pizza. So, you have 5 + 5 + 3 = 13 pieces of pizza, each of which is an equal fifth of a pizza. Since a fraction is the number of pieces we have over the number of equal pieces in a whole, we can write the mixed number as the improper fraction .

To convert a mixed number to an improper fraction:

• Multiply the denominator and the whole number part.
• Add that answer to the numerator to get the improper numerator (the denominator stays the same).

To convert an improper fraction to a mixed number:

• Divide the denominator into the numerator.
• The quotient is the whole number part; the remainder is the numerator.

Section 1.1.4: Adding and Subtracting Proper Fractions

The key to adding or subtracting fractions is to remember that you can only add together pieces that are the same size.

• If the denominators are not the same, then you have to change your fractions to equivalent fractions so that they have the same denominators. Thus you will be adding or subtracting pieces that are the same size.

• You can multiply the denominators together to get a common denominator, but you will do the least amount of work in the long run if you use the Least Common Denominator (LCD).

• The LCD has to have all the factors from every denominator:

Section 1.1.5: Adding and Subtracting Mixed Numbers

To add or subtract mixed numbers, first deal with the whole numbers, then deal with the fractions.

If the fractions add up to an improper fraction, convert the improper fraction to a mixed number, and then combine the whole numbers again.

If subtracting the fractions would result in a negative fraction, borrow from the whole number part to make the positive fraction bigger.

Section 1.1.6: Multiplying and Dividing Proper and Improper Fractions

Multiplying fractions:

The rule for multiplying proper (and improper) fractions is to take numerator times numerator and denominator times denominator. Example:

One way to be efficient is to cancel common factors before you multiply:

Dividing fractions: One way to help understand what happens when dividing with fractions:

• If you divide by a number that is bigger than the number one, then you get a smaller number, because you are dividing something into smaller pieces.

• If you divide by a number that is smaller than one, you get a bigger number, because when you divide something into smaller pieces, then you get more pieces overall.

The rule for dividing proper (and improper) fractions is to take the reciprocal of the second fraction, then multiply. Example:

Section 1.1.7: Multiplying and Dividing Mixed Numbers

To multiply or divide mixed numbers, you have to convert them to improper fractions, then use the regular rules for multiplying and dividing fractions just covered.

Section 1.2: Decimals

Section 1.2.1: The Decimal Representation (for Terminating Decimals)

• Positional notation as increasing factors of 10…
• Positional notation as decreasing factors of 10…  once get past 1  10, you’re in the decimal range…
• The positional notation can be used to write a fraction equivalent:

• Or you can use the trick of writing a fraction with the digits after the decimal in the numerator and the next larger power of 10 in the denominator:

Section 1.2.2: General Conversion between Fractions and Decimals

• To convert from fraction to a decimal, you can try to write the fraction as an equivalent fraction with a denominator that is a power of 10.
• Or you can do long division, remembering to put a decimal at the end if quotient and dividend. Stop when you get zero for a remainder…
• For some fractions, you never get a remainder of zero no matter how long you do the long division. These fractions have a decimal representation that repeats forever, and are called non-terminating or repeating decimals. You know you have a repeating decimal when the long-division pattern keeps on repeating.
• A bar over the repeating pattern is frequently used to represent a repeating decimal.

Section 1.2.3: Rounding Decimals

• A place to round to will be specified for you with phrases like “round to two decimal places” or “round to the nearest hundredth”. The idea is that you will re-write the number, but only up to that place, and with the last digit rounded according to the following rules:
• If the next digit is less than 5, leave the last digit alone
• If the next digit is 5 or more, add 1 to the last digit.

• For an example, here is how to round the number 10,547.395 to different specified decimal places:

Section 1.2.4: Adding and Subtracting Decimals

• To add: Line up the decimal points and add the digits from right to left, carrying as necessary.
• To subtract: Line up the decimal points and subtract the digits from right to left, borrowing as necessary.

Section 1.2.5: Multiplying Decimals

• Multiply as if there were no decimals present
• The answer will have as many decimal places as all the decimal places of both factors added up.

Section 1.2.6: Dividing Decimals

• Move the decimal point the same number of places in the divisor and dividend until the divisor is a whole number.
• Then do standard long division, putting a decimal point in the quotient right above the decimal point in the dividend.

Section 1.3: Significant Digits

Section 1.3.1: Identifying and Writing Significant Digits

• There is a shortcut that scientists and engineers use to state the precision of a measurement when writing down the measurement.
• For example, if you measure the width of a box of chalk with a school-type ruler, you might get an answer of 6.1 cm. Since the ruler is only marked in increments of 0.1 cm, you really can’t measure any finer than half that increment, which is 0.05 cm. Thus, the best way to state your measurement is as 6.1 cm  0.05 cm. That way, people know exactly how precise your measurement is, and they know not to expect more precision (like 6.139 cm), and that a number of 6 cm is neglecting the full precision of the measurement. Note: 6.1 cm  0.05 cm = 61 mm  0.5 mm…
• If you measure the width with a high-precision instrument like a digital caliper, then you might discover the answer to be 61.22 mm. Again, the uncertainty would be  one-half of the smallest measurement possible, which would be  0.005 mm. Thus, we’d write the measurement as 61.22 mm  0.005 mm.
• However, people have found a more efficient way to write a measurement like 6.1 cm  0.05 cm. Instead, they just write 6.1 cm, and it is understood that the precision is  one-half of the farthest-right digit. In this case, significant digits are being used to indicate the precision of the measurement.
• The significant digits (also called significant figures and abbreviated sig figs) of a number are those digits that carry meaning contributing to its precision.

• That last practice problem embodies one of the most confusing parts of working with significant digits: measurements that have a lot of zeros in them.
• When you want to show that a zero is significant, you go to the extra effort of writing extra stuff that you normally wouldn’t write, like putting extra zeros after the decimal point to show the finest increment of measurement:
• Example: 100.00 m
• Continuing the theme of writing something a little extra to show what zeros are significant, an old-fashioned trick for showing which trailing zeros are significant is to put a bar over every significant zero:
• Like or for the last two examples above.
• A more widely used trick to show when a measurement has been made to the nearest ones place is to put a decimal after the ones palce:

• The rule for identifying the significant digits in a measurement can be stated in two ways:
• All digits are significant except leading zeros after a decimal point and trailing zeros before it. Both types of zeros merely serve as placeholders.

• “point right, otherwise left”:
• If there is a decimal point present, find the left-most nonzero digit, and then count digits toward the right. If there is no decimal point in the number, find the right-most nonzero digit and count toward the left. In both cases, keep counting digits until you reach the other end of the number.

Section 1.3.2: Significant Digits after Adding or Subtracting

• When you add or subtract measurements, then you have to round your answer to the precision of the least accurate measurement. That’s because the uncertainty in the least precise measurement overwhelms the precision of the other measurements.

Section 1.3.3: Significant Digits after Multiplying or Dividing

• Rule: Find the measurement with the fewest significant figures. Round your answer to that many significant figures.
• One exception to the rule: numbers without a unit of measurement attached are considered exact numbers that have an infinite number of significant digits. Ignore them when determining the number of sig. figs. in your answer.
• Example: 100  22.85 cm = 2285 cm

Section 1.4: Signed Numbers

Section 1.4.1: The Concept of Negative Numbers

• To represent a change that decreases, we use negative numbers (positive numbers represent an increase)
• The number line is one convenient way to visualize positive and negative numbers:

• Positive (or “ordinary”) numbers are to the right of zero
• Negative numbers are to the left of zero.
• The number line could represent measurements like yards from the line of scrimmage, dollars in your checking account, or miles to the east or west of your current location.

• The first important concept of signed numbers is the “opposite” or “additive inverse” of a number.
• When a signed number and its opposite are added together, the answer is zero.
• The opposite of 5 is –5 since 5 + (-5) = 0
• For example, if you make $5 then lose $5, you’re back to zero dollars
• The opposite of –5 is 5
• For example, if a running back loses 5 yards on first down, then gains 5 yards on second down, then the team is back to where it started on the line of scrimmage.
• To find the opposite of a signed number, just change the sign that is in front of the number.

Section 1.4.2: Adding and Subtracting Signed Numbers

• To add a signed number to another number, move that amount in the positive direction for a positive number, or move that amount in the negative direction for a negative number.

• To subtract a signed number from another number, move that amount in the opposite direction of the sign of the subtracted number.

Section 1.4.3: Multiplying and Dividing Signed Numbers

• A positive number times a positive number has a positive answer. (A friend of a friend is a friend.)
• A positive number times a negative number has a negative answer. (A friend of an enemy is an enemy; an enemy of a friend is an enemy.)
• A negative number times a negative number has a positive answer. (An enemy of an enemy is a friend.)
• If I say “I am not dishonest”, the double negative makes the sentence equivalent to saying “I am honest”.
• The signs for dividing are the same as for multiplying, because a division problem can always be re-written as a multiplication problem:

Section 1.5: Exponents

Section 1.5.1 Definition of Positive Integer Exponents

Exponents were originally developed as a shortcut notation for repeated multiplication:

Section 1.5.5 Simplifying Expressions Involving Two or More of These Forms

Some expressions will involve more than one of the exponent laws. You will need to carefully determine which laws to apply. In general, start by simplifying within any parentheses groups. Work your way outward, applying exponents to the factors of each group, and then finish by multiplying the simplified groups together.

Section 1.5.6The Meaning of Zero and Negative Exponents

If we look at the pattern of repeatedly dividing a number that is a perfect power by the base, we can see a meaningful value for a base raised to a power of zero, and for a base raised to a negative power:

Any base (except a base of zero) raised to a power of zero is equal to one. Here is another explanation:

Any base raised to a negative power is the same as that base raised to the positive of that power, but on the other side of the fraction bar.

Section 1.5.10 Scientific Notation

• Exponents can be used to write very large and very small numbers in a more concise way, called scientific notation.
• Scientific notation is a way to write either a very large number or a number very close to zero using a number between 1 and 10 times a power of ten.
• Scientific notation is considered simpler because there is no need to write a long string of 0 digits at the end of a large number, or long string of 0’s in a decimal that is very close to zero.

WritingNumbers in Scientific Notation

 shortcut: count how many times you have to move the decimal to get it behind the first significant digit; that count is the positive power of ten for large numbers or the negative power of ten for numbers close to zero

Change numbers written in scientific notation to standard (decimal) form

 shortcut: move the decimal left or right by the power of ten. Positive exponents make a big number, negative exponents make a decimal close to zero.

Section 1.5.11 Multiplying and DividingNumbers Using Scientific Notation

Using scientific notation can simplify calculations involving large numbers. First, convert numbers to scientific notation, and then use exponent rules to reduce the powers of 10.

Section 1.6: Order of Operations

Order of Operations:

1.Perform all operations that appear in parentheses (or other grouping symbols) first. If grouping symbols are nested, do the innermost first.

2.Raise all bases to powers in the order encountered moving from left to right.

3.Perform all multiplications/divisions in the order encountered moving from left to right.

4.Perform all additions/subtractions in the order encountered moving from left to right.

Here, “other grouping symbols” means brackets [ ], braces { }, numerators or denominators of fractions, and radicals, like square roots.

An example of a nested expression is. The innermost grouping symbol is (4+1) so the result is

.

Note that expressions like 2(3 + 7) can be evaluated in two ways: using either the order of operations, or the “Distributive Property:” a(b + c) = ab + ac.

In addition to parentheses, brackets and braces, certain symbols act as implied grouping symbols.The most important of these are the fraction bar and the square root symbol.

The fraction bar acts to separate the numerator from the denominator. If either or both of the numerator or denominator consist of an expression with operations, these must be performed first before the division indicated by the fraction bar. For example,