Civil Service Exam: Mathematics Basics

A person with a well-rounded education should be able to perform all but the most complicated math operations either mentally or with pencil and paper.

This is why civil service exams have math questions. An office worker shouldn't need to stop and open up the computer's calculator function every time he needs to add, subtract, multiply or divide. He/she should be able to do basic math mentally. So, in order to do well on your civil service exam, you'll need to make sure your math knowledge and skills are in good shape. If you've been out of school for several years, your skills might have gotten a little rusty, and you may not be as familiar with some of the more advanced or less frequently used concepts as you once were.

This guide is a thorough refresher course in math to get you up to speed quickly. It starts with the basics and then builds on that foundation all the way through algebra and geometry. (Yes, you may very well encounter basic algebra or geometry questions, or both, on the civil service test.) Even if you struggled with math in school, you'll find all the information you need to master the math portion of the civil service exam in this guide. Take your time and go at your own pace, and use the practice test at the end as a diagnostic tool to measure your readiness for the exam.

Numbers and Their Classifications

Numbers are the basic building blocks of mathematics. Specific features of numbers are identified by the following terms:

Integers—the set of whole positive and negative numbers, including zero. Integers do not include fractions (
), decimals

(0.56), or mixed numbers (
).

Prime number—a whole number greater than 1 that has only two factors, itself and 1. An example is 11, which is only divisible by 1 and 11.

However, 1 itself is not a prime number since it has only one factor, 1.

Composite number—a whole number greater than 1 that has more than two factors; in other words, any whole number that is not a prime number. For example, the composite number 8 has the factors of 1, 2, 4, and 8.

Even number—any integer that can be divided by 2 without leaving a remainder. The integers 2, 4, 6, 8, and so on are even numbers.

Odd number—any integer that cannot be divided evenly by 2. The integers 3, 5, 7, 9, and so on are odd numbers.

Decimal number—a non-integer that uses a decimal point to show the part of the number that is less than one. An example of a decimal number is


1.234.

Decimal point—a symbol used to separate the ones place from the tenths place in decimals or dollars from cents in currency.

Decimal place—the position of a number to the right of the decimal point. In the decimal 0.123, the 1 is in the first place to the right of the decimal point, indicating tenths; the 2 is in the second place, indicating hundredths; and the 3 is in the third place, indicating thousandths.

The decimal, or base 10, system is a number system that uses ten different digits (
). An example of a number system that uses something other than ten digits is the binary, or base 2, number system, used by computers, which uses only the numbers 0 and 1.

It is thought that the decimal system originated because people used their 10 fingers for counting.

Rational, irrational, and real numbers can be described as follows:

Rational numbers include all integers, decimals, and fractions.

Any terminating or repeating decimal number is a rational number.

Irrational numbers cannot be written as fractions or decimals because the number of decimal places is infinite and there is no recurring pattern of digits within the number. For example, pi (π) begins with 3.141592 and continues without terminating or repeating, so pi is an irrational number.

Real numbers are the set of all rational and irrational numbers.


Operations

There are four basic mathematical operations:· Addition increases the value of one quantity by the value of another quantity. For example,
.
The result is called the sum. With addition, the order does not matter:
.


Subtraction is the opposite operation to addition; it decreases the value of one quantity by the value of another quantity. For example,
.


The result is called the difference. With subtraction, the order does matter:
.


Multiplication can be thought of as repeated addition. One number tells how many times to add the other number to itself. For example,
.
With multiplication, the order does not matter:
 or
.


Division is the opposite operation to multiplication; one number tells us how many parts to divide the other number into. For example,
;

20 can be split into 4 equal parts of 5. With division, the order of the numbers does matter:
.



Exponents
An exponent is a superscript number placed at the top right of another number. It indicates how many times the base number is to be multiplied by itself. Exponents provide a shorthand way to write what would be a longer mathematical expression. For example,
;
.
A number with an exponent of 2 is said to be 'squared,' while a number with an exponent of 3 is said to be 'cubed.' The value of a number raised to an exponent is called its power. So,
 is read as '8 to the 4th power,' or '8 raised to the power of 4.' A negative exponent is the same as the reciprocal of a positive exponent. For example,
.


The laws of exponents are as follows:

·     

Any number to the power of 1 is equal to itself:
.


· The number 1 raised to any power is 1:
.


· Any number raised to the power of 0 is 1:
.


Add exponents to multiply powers of the same base number:
.


Subtract exponents to divide powers of the same number:
.


Multiply exponents to raise a power to a power:
.


If multiplied or divided numbers inside parentheses are collectively raised to a power, this is the same as each individual term being raised to that power:
;
.


Exponents do not have to be integers. Fractional or decimal exponents follow all the rules above as well. Example:
.


A root, such as a square root, is another way of writing a fractional exponent. Instead of using a superscript, roots use the radical symbol (
) to indicate the operation. A radical has a number underneath the bar, and may sometimes also have a number in the upper left:
, read as 'the

nth root of a.' The relationship between radical notation and exponent notation can be described by this equation:
.
The two special cases of n = 2 and n = 3 are called square roots and cube roots. If there is no number to the upper left, it is understood to be a square root (
). Nearly all of the roots you encounter will be square roots. A square root is the same as a number raised to the one-half power. When we say that a is the square root of b (
), we mean that a multiplied by itself equals b: (
).


Parentheses

Parentheses are used to designate which operations should be done first in equations with multiple operations. For example, 4 – (2 + 1) = 1; the parentheses tell us that we must first add 2 and 1, and then subtract the sum from 4, rather than simply moving from left to right: 4 – 2 + 1 = 3.


Order of Operations

Order of operations is a set of rules that dictates the order in which we must perform each operation in an expression. If an expression includes multiple different operations, the order of operations tells us which to do first. The best way to remember the order of operations is to use the acronym PEMDAS, or 'Please Excuse My Dear Aunt Sally.' PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It is important to understand that multiplication and division have equal precedence, as do addition and subtraction, so those pairs of operations are simply worked from left to right in order.

Example: Evaluate the expression
 using the correct order of operations.

P—perform the operations inside the parentheses:
.


The equation now looks like this:
.


E—simplify the exponents:
.


The equation now looks like this:
.


MD—perform multiplication and division from left to right:
;
.


The equation now looks like this:
.


AS—perform addition and subtraction from left to right:
;
.



Perfect Square

A perfect square is a number that has an integer for its square root. There are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the squares of integers 1 through 10).


Scientific Notation

Scientific notation is a way of writing large numbers in a shorter form. The form
 is used in scientific notation, where a is greater than or equal to 1, but less than 10, and n is the number of places the decimal must move to get from the original number to a. For example, the number 230,400,000 is cumbersome to write. To write the value in scientific notation, place a decimal point between the first and second numbers, and include all digits through the last nonzero digit (
). To find the appropriate power of 10, count the number of places the decimal point had to move (
). The number is positive if the decimal moved to the left, and negative if it moved to the right. We can then write 230,400,000 as
.
If we look instead at the number 0.00002304, we have the same value for a, but this time the decimal moved 5 places to the right (
).

Thus, 0.00002304 can be written as
.
Using this notation makes it simple to compare very large or very small numbers. By comparing exponents, it is easy to see that
 is smaller than
, because 4 is less than 5.


Factors and Multiples

Factors are numbers that are multiplied together to obtain a product. For example, in the equation
, the numbers

2 and 3 are factors and 6 is the product. A prime number has only two factors

(1 and itself), but other numbers can have many factors. A common factor is a number that can divide into two or more other numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 15 are 1, 3, 5, and 15.

The common factors of 12 and 15 are 1 and 3. A prime factor is a prime number that divides into another number. The prime factors of 12 are 2 and 3. For 15, the prime factors are 3 and 5. The greatest common factor (GCF) is the largest number that is a factor of two or more numbers. For example, the factors of 15 are 1, 3, 5, and 15; the factors of 35 are 1, 5, 7, and 35. Therefore, the greatest common factor of 15 and 35 is 5. The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the multiples of 3 include 3, 6, 9, 12, 15, etc.; the multiples of 5 include 5, 10, 15, 20, etc. Therefore, the least common multiple of 3 and 5 is 15.


Fractions, Percentages, and Related Concepts


Fractions

A fraction is a number expressed as one integer written above another integer, with a dividing line between them
.


It represents the quotient of the two numbers: x divided by y. It can also be thought of as x out of y equal parts. The top number of a fraction is called the numerator, and it represents the number of parts under consideration. The 1 in
 means that 1 part out of the whole is being considered in the calculation. The bottom number of a fraction is called the denominator, and it represents the total number of equal parts. The 4 in
 means that the whole consists of 4 equal parts. A fraction cannot have a denominator of zero; this is referred to as 'undefined.'

Fractions can be manipulated without changing their value by multiplying or dividing (but not adding or subtracting) both the numerator and denominator by the same number. For example,
.
If you divide both numbers by a common factor, you are reducing or simplifying the fraction. Two fractions that have the same value but are expressed differently are known as equivalent fractions. For example,
 are all equivalent fractions. They can also all be reduced or simplified to
.


When two fractions are manipulated so that they have the same denominator, this is known as finding a common denominator. The number chosen to be that common denominator should be the least common multiple of the two original denominators. For example, if you want to add
 you must first find the common denominator. The least common multiple of 4 and 6 is 12.

Manipulating to achieve the common denominator:
.


If two fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two numerators and retaining the same denominator. If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted. For example:
.


Two fractions can be multiplied by multiplying the two numerators to find the new numerator and the two denominators to find the new denominator. For example:
.


Two fractions can be divided by flipping the numerator and denominator of the second fraction and then proceeding as though it were a multiplication problem. For example:
.


A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction. Proper fractions have values less than one and improper fractions have values greater than one.


Mixed Numbers

A mixed number contains both an integer and a fraction. Any improper fraction can be rewritten as a mixed number. For example,
.
Similarly, any mixed number can be rewritten as an improper fraction. For example:
.



Percentages

Percentages can be thought of as fractions that are based on a whole of 100 instead of 1; that is, one whole is equal to 100%. The word percent means 'per hundred.' Fractions can be expressed as percentages by finding equivalent fractions with a denomination of 100. For example:
;
.


To express a percentage as a fraction, divide by 100 and reduce the fraction to its simplest possible terms. For example:
;
.


Converting decimals to percentages and percentages to decimals is as simple as moving the decimal point. To convert from a decimal to a percent, move the decimal point two places to the right. To convert from a percent to a decimal, move it two places to the left. For example, 0.23 = 23%;


5.34 = 534%; 0.007 = 0.7%; 700% = 7.00; 86% = 0.86; 0.15% = 0.0015.

It may be helpful to remember that the percentage number will always be greater than the equivalent decimal number.

A percentage problem can be presented three main ways: (1)

Find what percentage of some number another number is. Example: What percentage of 40 is 8? (2) Find what number is some percentage of a given number. Example:

What number is 20% of 40? (3) Find what number another number is a given percentage of. Example: 8 is 20% of what number? The three components in all of these cases are the same: a whole (W), a part (P), and a percentage

(%). These are related by the following equation:
.
This is the form of the equation you would use to solve problems of type (2). To solve types (1) and (3), you would use these two forms:
 and
.


The thing that frequently makes percentage problems difficult is that they are typically word problems, so a large part of solving them is figuring out which quantities are W, P, and %. For example: In a school cafeteria, 7 students choose pizza, 9 choose hamburgers, and 4 choose tacos. Find the percentage that chooses tacos. To find the whole, you must first add all of the parts: 7 + 9 + 4 = 20. The percentage can then be found by dividing the part by the whole:
 ;
.



Ratio

A ratio is a comparison of two quantities in a particular order. For example: If there are 14 computers in a lab, and the class has 20 students, there is a student to computer ratio of 20 to 14, commonly written as 20:14. Ratios are normally reduced to their smallest whole number representation, so 20:14 would be reduced to 10:7 by dividing both sides by 2.


Proportion

A proportion is a relationship between two quantities that dictates how one changes when the other changes. A direct proportion describes a relationship in which a quantity increases by a set multiple for every increase in the other quantity, or decreases by that same amount for every decrease in the other quantity. Example: Assuming a constant driving speed, the time required for a car trip increases as the distance of the trip increases.

The distance to be traveled and the time required to travel are directly proportional.

Inverse proportion is a relationship in which an increase in one quantity is accompanied by a decrease in the other, or vice versa. Example: the time required for a car trip decreases as the speed increases, and increases as the speed decreases, so the time required is inversely proportional to the speed of the car.


Data Analysis


Statistics

Statistics is the branch of mathematics that deals with collecting, recording, interpreting, illustrating, and analyzing large amounts of data. The following terms are often used in the discussion of data and statistics:

Data—the collective name for pieces of information (singular is datum).

Quantitative data—measurements (such as length, mass, and speed) that provide information about quantities in numbers.

Qualitative data—information (such as colors, scents, tastes, and shapes) that cannot be measured using numbers.

Discrete data—information that can be expressed only by a specific value, such as whole or half numbers. For example, since people can be counted only in whole numbers, a population count would be discrete data.

Continuous data—information (such as time and temperature) that can be expressed by any value within a given range.

Primary data—information that has been collected directly from a survey, investigation, or experiment, such as a questionnaire or the recording of daily temperatures. Primary data that has not yet been organized or analyzed is called raw data.

Secondary data—information that has been collected, sorted, and processed by the researcher.

Ordinal data—information that can be placed in numerical order, such as age or weight.

Nominal data—information that cannot be placed in numerical order, such as names or places.


Measures of Central Tendency

The quantities of mean, median, and mode are all referred to as measures of central tendency. They can each give a picture of what the whole set of data looks like with just a single number. Knowing what each of these values represents is vital to making use of the information they provide. The mean, also known as the arithmetic mean or average, of a data set is calculated by summing all of the values in the set and dividing that sum by the number of values. For example, if a data set has 6 numbers and the sum of those 6 numbers is 30, the mean is calculated as 30 ÷ 6 = 5. The median is the middle value of a data set when all values are in numerical order. In the data set (1, 2, 3, 4, 5), the median is 3. If there is an even number of values in the set, the median is calculated by taking the average of the two middle values. In the data set (1, 2, 3, 4, 5, 6), the median would be (3 + 4) ÷ 2 = 3.5. The mode is the value that appears most frequently in the data set. In the data set (1, 2, 3, 4, 5, 5, 5), the mode would be 5 since this value appears three times. If multiple values appear the same number of times, there are multiple values for the mode. If the data set were (1, 2, 2, 3, 4, 4, 5, 5), the modes would be 2, 4, and 5. If no value appears more than any other value in the data set, then there is no mode.


Measures of Dispersion

The standard deviation expresses how spread out the values of a distribution are from the mean. Standard deviation is given in the same units as the original data and is represented by a lower-case sigma (σ). A high standard deviation means that the values are very spread out. A low standard deviation means that the values are close together. If every value in a distribution is increased or decreased by the same amount, the mean, median, and mode are increased or decreased by that amount, but the standard deviation stays the same. If every value in a distribution is multiplied or divided by the same number, the mean, median, mode, and standard deviation are all multiplied or divided by that number.


Range of Distribution

The range of a distribution is the difference between the highest and lowest values in the distribution. For example, in the data set (1, 3, 5, 7, 9, 11), the highest and lowest values are 11 and 1, respectively. The range would then be calculated: 11 – 1 = 10.


Three Quartiles

The three quartiles are the three values that divide a data set into four equal parts. Quartiles are generally only calculated for data sets with a large number of values. As a simple example, for the data set consisting of the numbers 1 through 99, the first quartile (Q1) would be 25, the second quartile (Q2), always equal to the median, would be 50, and the third quartile (Q3) would be 75. The difference between Q1 and Q3 is known as the interquartile range.


Probability

Probability is a branch of statistics that deals with the likelihood of something taking place. One classic example is a coin toss. There are only two possible results: heads or tails. The likelihood, or probability, that the coin will land as heads is 1 out of 2 (1/2, 0.5, 50%). Tails has the same probability. Another common example is a 6-sided die roll. There are six possible results from rolling a single die, each with an equal chance of happening, so the probability of any given number coming up is 1 out of 6.

Terms frequently used in probability:

Event—a situation that produces results of some sort (such as a coin toss).

Compound event—an event that involves two or more items (rolling a pair of dice; taking the sum).

Outcome—a possible result in an experiment or event (heads, tails).

Desired outcome (or success)—an outcome that meets a particular set of criteria (a roll of 1 or 2 if a number less than 3 is needed).

Independent events—two or more events whose outcomes do not affect one another (two coins tossed at the same time).

Dependent events—two or more events whose outcomes affect one another (two cards drawn consecutively from the same deck).

Certain outcome—probability of outcome is 100% or 1.

Impossible outcome—probability of outcome is 0% or 0.

Mutually exclusive outcomes—two or more outcomes whose criteria cannot all be satisfied in a single event (a coin coming up heads and tails on the same toss).

Probability is the likelihood of a certain outcome for a given event. The theoretical probability can usually be determined without actually performing the event. The probability of an outcome occurring is given by the formula:


where P(A) is the probability of an outcome A occurring, and each outcome is just as likely to occur as every other outcome.

If each outcome has the same probability as every other possible outcome, the outcomes are said to be equally likely to occur. The total number of acceptable outcomes must be less than or equal to the total number of possible outcomes.

If the two are equal, then the outcome is certain to occur and the probability is 1. If the number of acceptable outcomes is zero, then the outcome is impossible and the probability is 0. For example: There are 20 marbles in a bag and 5 are red. The theoretical probability of randomly selecting a red marble is 5 out of 20, (5/20 = 1/4, 0.25, or 25%).


Permutations and Combinations

When trying to calculate the probability of an event using the (desired outcomes)/(total outcomes) formula, you may frequently find that there are too many outcomes to individually count them. Permutation and combination formulas offer a shortcut to counting outcomes. The primary distinction between permutations and combinations is that permutations take into account order, while combinations do not. To calculate the number of possible groupings, there are two necessary parameters: the number of items available for selection and the number to be selected. The number of permutations of r items given a set of n items can be calculated as
.
The number of combinations of r items given a set of n items can be calculated as
 or
.
For example: Suppose you want to calculate how many different 5-card hands can be drawn from a deck of 52 cards. This is a combination since the order of the cards in a hand does not matter. There are 52 cards available, and 5 to be selected. Thus, the number of different possible hands is
.



Complement of an Event

Sometimes it may be easier to calculate the possibility of something not happening, or the complement of an event. Represented by the symbol
, the complement of A is the probability that event A does not happen.

When you know the probability of event A occurring, you can use the formula

, where
 is the probability of event A not occurring, and
 is the probability of event A occurring.


Addition Rule

The addition rule for probability is used for finding the probability of a compound event. Use the formula
P(A) is the probability of the event A occurring, P(B) is the probability of event B occurring, and P(A and B) is the probability of both events occurring to find the probability of a compound event. The probability of both events occurring at the same time must be subtracted to eliminate any overlap in the first two probabilities.


Conditional Probability

Conditional probability is the probability of a dependent event occurring once the original event has already occurred. Given event A and dependent event B, the probability of event B occurring when event A has already occurred is represented by the notation
 because there is one ball with a 7 on it and 10 balls to choose from. Assuming the first draw did not yield a 7, the probability of drawing a 7 on the second draw is now
 because there are only 9 balls remaining for the second draw.


Multiplication Rule

The multiplication rule can be used to find the probability of two independent events occurring using the formula P(A and B)= P(A) P(B), where P(A and B) is the probability of two independent events occurring, P(A) is the probability of the first event occurring, and P(B) is the probability of the second event occurring.

The multiplication rule can also be used to find the probability of two dependent events occurring using the formula

P(A and B) is the probability of two dependent events occurring, P(A) is the probability of the first event occurring, and
.
For example, to find the probability that at least one even number will show when a pair of dice is rolled, find the probability that two odd numbers will be rolled (no even numbers) and subtract from one. You can always use a tree diagram or make a chart to list the possible outcomes when the sample space is small, such as in the dice-rolling example, but in most cases, it will be much faster to use the multiplication and complement formulas.


Expected Value

Expected value is a method of determining the expected outcome in a random situation. It is really a sum of the weighted probabilities of the possible outcomes. Multiply the probability of an event occurring by the weight assigned to that probability (such as the amount of money won or lost).

A practical application of the expected value is to determine whether a game of chance is really fair. If the sum of the weighted probabilities is equal to zero, the game is generally considered fair because the player has a fair chance to at least to break even. If the expected value is less than zero, then players lose more than they win. For example, a lottery drawing might allow the player to choose any three-digit number, 000–999. The probability of choosing the winning number is 1:1,000. If it costs $1 to play, and a winning number receives $500, the expected value is
.
In other words, you can expect to lose, on average, 50 cents for every dollar you spend.

Experimental Probability

Most of the time, when we talk about probability, we mean theoretical probability. Experimental probability, or empirical probability or relative frequency, is the number of times an outcome occurs in a particular experiment or a certain number of observed events. While theoretical probability is based on what should happen, experimental probability is based on what has happened. Experimental probability is calculated in the same way as theoretical, except that actual outcomes are used instead of possible outcomes.

Theoretical and experimental probability do not always line up with one another. Theoretical probability says that out of 20 coin tosses, 10 should be heads. However, if we were actually to toss 20 coins, we might record just 5 heads. This doesn't mean that our theoretical probability is incorrect; it just means that this particular experiment had different results from what was predicted. A practical application of empirical probability is the insurance industry. There are no set functions that define life span, health, or safety. Insurance companies look at factors from hundreds of thousands of individuals to find patterns that they then use to set the formulas for insurance premiums.

Objective Probability

Objective probability is based on mathematical formulas and documented evidence. Examples of objective probability include raffles or lottery drawings with a predetermined number of possible outcomes and a predetermined number of outcomes that correspond to an event. Other cases of objective probability include probabilities of rolling dice, flipping coins, or drawing cards. Most gambling games are based on objective probability.


Subjective Probability

Subjective probability is based on personal or professional feelings and judgments. Often, there is a lot of guesswork following extensive research. Areas where subjective probability is applicable include sales trends and business expenses. Attractions set admission prices based on subjective probabilities of attendance, based on varying admission rates, in an effort to maximize their profit.


Common Charts and Graphs


Bar Graph, Line Graph, and Pictograph

A bar graph uses bars to compare data, as if each bar were a ruler being used to measure the data. The graph includes a scale that identifies the units being measured. A line graph connects points to show how data increases or decreases over time. The time line is the horizontal axis. The connecting lines between data points on the graph are a way to more clearly show how the data changes. The image below is a line graph.


A pictograph uses pictures or symbols to show data. The pictograph has a key to identify what each symbol represents. Generally, each symbol stands for one or more objects.

A pie chart or circle graph is a diagram used to compare parts of a whole. The full pie represents the whole, and it is divided into sectors that each represent a part of the whole.

Each sector or slice of the pie is either labeled to indicate what it represents, or explained on a key associated with the chart. The size of each slice is determined by its percentage of the whole. Numerically, the angle measurement of each sector can be computed by solving the proportion: x/360 = part/whole.



Histogram

A histogram is a special type of bar graph where the data are grouped in intervals (for example 20–29, 30–39, 40–49, etc.). The frequency, or number of times a value occurs in each interval, is indicated by the height of the bar. The intervals do not have to be the same amount but usually are (all data in ranges of 10 or all in ranges of 5, for example). The smaller the intervals, the more detailed the



Stem-and-Leaf Plot

A stem-and-leaf plot is a way to organize data visually so that the information is easy to understand. A stem-and-leaf plot is simple to construct because a simple line separates the stem (the part of the plot listing the tens digit, if displaying two-digit data) from the leaf (the part that shows the ones digit). Thus, the number 45 would appear as 4 │ 5. The stem-and-leaf plot for test scores of a group of 11 students might look like the following:

9 │ 5

8 │ 1, 3, 8

7 │ 0, 2, 4, 6, 7

6 │ 2, 8

A stem-and-leaf plot is similar to a histogram or other frequency plot, but with a stem-and-leaf plot, all the original data is preserved. In this example, while all data has been maintained, it can be seen at a glance that nearly half the students scored in the 70's. These plots can be used for larger numbers as well, but they tend to work better for small sets of data as they can become unwieldy with larger sets.


Equations and Graphing

When algebraic functions and equations are shown graphically, they are usually shown on a Cartesian

Coordinate Plane. This consists of two number lines placed perpendicular to each other and intersecting at the zero point, also known as the origin. The horizontal number line is known as the x-axis, with positive values to the right of the origin and negative values to the left of the origin. The vertical number line is known as the y-axis, with positive values above the origin and negative values below the origin. Any point on the plane can be identified by an ordered pair in the form (x,y), called coordinates. The x-value of the coordinate is called the abscissa, and the y-value of the coordinate is called the ordinate. The two number lines divide the plane into four quadrants: I, II, III, and IV.



Coordinate Plane and Graphing

Before learning the different forms of equations, it is important to understand some terminology. A ratio of the change in the vertical distance to the change in horizontal distance is called the slope. On a graph with two points,
 and
, the slope is represented by the formula
 ;
.
If the value of the slope is positive, the line slopes upward from left to right. If the value of the slope is negative, the line slopes downward from left to right. If the y-coordinates are the same for both points, the slope is 0 and the line is horizontal. If the x-coordinates are the same for both points, there is no slope and the line is vertical. Two or more lines with equal slopes are parallel. Perpendicular lines have slopes that are negative reciprocals of each other, such as
 and
.


Equations are made up of monomials and polynomials. A monomial is a single variable or product of constants and variables, such as x, 2x, or
 . There will never be addition or subtraction symbols in a monomial. Like monomials have like variables, but they may have different coefficients. Polynomials are algebraic expressions that use addition and subtraction to combine two or more monomials. Two terms make a binomial, three terms make a trinomial, etc. The degree of a monomial is the sum of the exponents of the variables. The degree of a polynomial is the highest degree of any individual term. A. mentioned previously, equations can be written many ways. Below is a list of the many forms equations can take.

Standard Form:
y-intercept is
m is the slope and b is the y-intercept.

Point-Slope Form:
 is a point on the line.

Two-Point Form:
 , where
 and
 are two points on the given line.

·     

Intercept Form:
, where
 is the point at which a line intersects the x-axis, and
 is the point at which the same line intersects the y-axis.

Equations can also be written as
, where
.
These are referred to as One Variable Linear Equations. A solution to such an equation is called a root. For example, in the equation
, if we solve for
 we get a solution of
.
In other words, the root of the equation is –2. This is found by first subtracting 10 from both sides, which gives
.
Next, simply divide both sides by the coefficient of the variable, in this case 5, to get
.
This can be checked by plugging –2 back into the original equation
.

 

Geometry


Lines and Planes


Point, Line, and Plane

A point is a fixed location in space with no size or dimensions. It is commonly represented by a dot. A line is a set of points that extends infinitely in two opposite directions. It has length, but no width or depth. A line can be defined by any two distinct points. A line segment is a portion of a line with definite endpoints. A ray is a portion of a line that extends from a single point on that line in one direction along the line. It has a definite beginning, but no ending. A plane is a two-dimensional flat surface defined by three noncollinear points. A plane extends an infinite distance in all directions in those two dimensions. It contains an infinite number of points, parallel lines and segments, intersecting lines and segments, as well as parallel or intersecting rays. A plane will never contain a three-dimensional figure or skew lines. Two given planes will either be parallel or will intersect to form a line. A plane may intersect a circular conic surface such as a cone to form conic sections such as the parabola, hyperbola, circle, or ellipse.


Perpendicular Lines, Parallel Lines, and Bisectors

Perpendicular lines intersect each other at right angles.

They are represented by the symbol
.
The shortest distance from a line to a point not on the line is a perpendicular segment from the point to the line. Parallel lines lie in the same plane. They have no points in common and never meet. It is possible for lines to be in different planes, have no points in common, and never meet, but not be parallel because they are in different planes. A bisector is a line or segment that divides another segment into two equal lengths. A perpendicular bisector of a line segment is composed of points that are equidistant from the endpoints of the segment it is dividing.


Intersecting Lines, Concurrent Lines, and Transversals

Intersecting lines have exactly one point in common.

Concurrent lines are multiple lines that intersect at a single point. A transversal is a line that intersects at least two other lines, which may or may not be parallel to one another. A transversal that intersects parallel lines is a common occurrence in geometry.


Angles
An angle is formed when two lines or line segments meet at a common point. It may be a common starting point for a pair of segments or rays, or it may be the intersection of lines. Angles are represented by the symbol ∠. The vertex is the point at which two segments or rays meet to form an angle. If the angle is formed by intersecting rays, lines, and/or line segments, the vertex is the point at which four angles are formed. The pairs of angles opposite one another are called vertical angles, and their measures are equal.

There are various types of angles: An acute angle has a degree measure less than 90°.

A right angle has a degree measure of exactly 90°. A. obtuse angle has a degree measure greater than 90° but less than 180°.

A straight angle has a degree measure of exactly 180°. This is also a semicircle.

A reflex angle has a degree measure greater than 180° but less than 360°.

A full angle has a degree measure of exactly 360°.

Two angles whose sum is exactly 90° are said to be complementary angles. The two angles may or may not be adjacent. In a right triangle, the two acute angles are complementary. Two angles whose sum is exactly 180° are said to be supplementary angles. The two angles may or may not be adjacent. Two intersecting lines always form two pairs of supplementary angles. Adjacent supplementary angles will always form a straight line. Two angles that have the same vertex and share a side are said to be adjacent. Vertical angles are not adjacent because they share a vertex but no common side.
          


When two parallel lines are cut by a transversal, the angles between the two parallel lines are interior angles and the angles that are outside the parallel lines are exterior angles. In the diagram below, angles 3, 4, 5, and 6 are interior angles. Angles 1, 2, 7, and 8 are exterior angles. that are in the same position relative to the transversal and a parallel line are corresponding angles. The diagram below has four pairs of corresponding angles: angles 1 and 5, angles 2 and 6, angles 3 and 7, and angles 4 and 8.

Corresponding angles formed by parallel lines are congruent.

When two parallel lines are cut by a transversal, the two interior angles on opposite sides of the transversal are called alternate interior angles. In the diagram below, there are two pairs of alternate interior angles: angles 3 and 6, and angles 4 and 5.

Alternate interior angles formed by parallel lines are congruent. When two parallel lines are cut by a transversal, the two exterior angles that are on opposite sides of the transversal are called alternate exterior angles. In the diagram below, there are two pairs of alternate exterior angles: angles 1 and 8, and angles 2 and 7. Alternate exterior angles formed by parallel lines are congruent. Congruent angles are represented by the symbol



When two lines intersect, four angles are formed. The nonadjacent angles at this vertex are called vertical angles. Vertical angles are congruent. In the diagram,
 and
.




Triangles

An equilateral triangle has three congruent sides. An equilateral triangle also has three congruent angles, 60° each. All equilateral triangles are also acute triangles since each angle is less than 90°.

An isosceles triangle has two congruent sides. An isosceles triangle also has two congruent angles opposite the two congruent sides.


A scalene triangle has no congruent sides. A scalene triangle also has three angles of different measures. The angle with the largest measure is opposite the longest side, and the angle with the smallest measure is opposite the shortest side.

An acute triangle is a triangle with all three angles less than 90°. If two of the angles are equal, the acute triangle is also an isosceles triangle. If the three angles are all equal, the acute triangle is also an equilateral triangle.

A right triangle has exactly one angle equal to 90°. All right triangles follow the Pythagorean Theorem
.
A right triangle can never be acute or obtuse. A. obtuse triangle has exactly one angle greater than 90°.

The other two angles may or may not be equal. If the two remaining angles are equal, the obtuse triangle is also an isosceles triangle.

Triangle Terminology

Altitude—a line segment drawn from one vertex perpendicular to the opposite side. In the diagram below,
,
, and
 are altitudes. The three altitudes in a triangle are always concurrent.


Height—the length of the altitude, although the two terms are often used interchangeably.

Orthocenter—the point of concurrency of the altitudes of a triangle. Note that in an obtuse triangle, the orthocenter will be outside the triangle, and in a right triangle, the orthocenter is the vertex of the right angle.

Median—a line segment drawn from one vertex to the midpoint of the opposite side. This is not necessarily the same as the altitude, except the altitude to the base of an isosceles triangle and all three altitudes of an equilateral triangle.

Centroid—the point of concurrency of the medians of a triangle.

Only in an equilateral triangle is this the same point as the orthocenter.

Unlike the orthocenter, the centroid is always inside the triangle. The centroid can also be considered the exact center of the triangle. Any shape of triangle can be perfectly balanced on a tip placed at the centroid. The centroid is also two-thirds of the distance from the vertex to the opposite side.

Pythagorean Theorem

In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called the legs. The Pythagorean Theorem states the relationship between the legs and hypotenuse of a right triangle:
a and b are the lengths of the legs and c is the length of the hypotenuse. Note that this formula will only work with right triangles.
s is any side length, since all three sides are the same length.
.
For many triangles, it may be difficult to calculate h, so using one of the other formulas given here may be easier.

Another formula that works for any triangle is
A is the area, s is the semiperimeter
A is the area and s is the length of a side. You could also use the 30° -

60° - 90° ratios to find the height of the triangle and then use the standard triangle area formula, but this is faster.

The area of an isosceles triangle can found by the formula


Polygons

Each straight line segment of a polygon is called a side. The point at which two sides of a polygon intersect is called the vertex. In a polygon, the number of sides is always equal to the number of vertices. A polygon with all sides congruent and all angles equal is called a regular polygon. A line segment from the center of a polygon, perpendicular to a side of the polygon, is called the apothem. In a regular polygon, the apothem can be used to find the area of the polygon using the formula
a is the apothem and p is the perimeter. A line segment from the center of a polygon to a vertex of the polygon is called a radius. The radius of a regular polygon is also the radius of a circle that can be circumscribed about the polygon.

Triangle—3-sided polygon

Quadrilateral—4-sided polygon

Pentagon—5-sided polygon

Hexagon—6-sided polygon

Heptagon—7-sided polygon

Octagon—8-sided polygon

Nonagon—9-sided polygon

Decagon—10-sided polygon

Dodecagon—12-sided polygon

More generally, an n-gon is a polygon that has n angles and n sides.

The sum of the interior angles of an n-sided polygon is (n – 2)180°. For example, in a triangle n = 3, so the sum of the interior angles is (3 – 2)180° = 180°. In a quadrilateral, n = 4, and the sum of the angles is (4 – 2)180° = 360°. The sum of the interior angles of a polygon is equal to the sum of the interior angles of any other polygon with the same number of sides.

A diagonal is a line segment that joins two nonadjacent vertices of a polygon.

A convex polygon is a polygon whose diagonals all lie within the interior of the polygon.

A concave polygon has at least one diagonal that lies outside the polygon. In the diagram below, quadrilateral ABCD is concave because diagonal


The number of diagonals in a polygon can be found by using the formula:
n is the number of sides in the polygon. This formula works for all polygons, not just regular polygons.
.



Similar figures are geometric figures that have the same shape, but do not necessarily have the same size.

All corresponding angles are equal, and all corresponding sides are proportional, but they do not have to be equal. Similar figures are indicated by the symbol
.



Note that all congruent figures are also similar, but not all similar figures are congruent.

A line of symmetry divides a figure or object into two symmetric parts. Each symmetric half is congruent to the other. An object may have no lines of symmetry, one line of symmetry, or more than one line of symmetry.

No lines of symmetry

One line of Multiple lines of symmetry


Quadrilaterals

A quadrilateral is a closed, two-dimensional geometric figure composed of exactly four straight sides. The sum of the interior angles of any quadrilateral is
.
A quadrilateral whose diagonals bisect each other is a parallelogram. A quadrilateral whose opposite sides are parallel (2 pairs of parallel sides) is a parallelogram. A quadrilateral whose diagonals are perpendicular bisectors of each other is a rhombus. A quadrilateral whose opposite sides (both pairs) are parallel and congruent is a rhombus. A parallelogram with right angles is a rectangle. A rhombus with right angles is a square. Because the rhombus is a form of parallelogram, the rules about the angles of a parallelogram also apply to the rhombus.

Parallelogram

A parallelogram is a quadrilateral with exactly two pairs of opposite parallel sides. The parallel sides are also congruent. The opposite interior angles are always congruent, and the consecutive interior angles are supplementary. The diagonals of a parallelogram bisect each other. Each diagonal divides the parallelogram into two congruent


Trapezoid

Traditionally, a trapezoid is a quadrilateral that has exactly one pair of parallel sides. Some math texts define a trapezoid as a quadrilateral with at least one pair of parallel sides.

Because there are no rules governing the second pair of sides, there are no rules that apply to the properties of the diagonals of a trapezoid.


Rectangles, Rhombuses, and Squares

Rectangles, rhombuses, and squares are all special forms of parallelograms. A rectangle is a parallelogram with right angles. All rectangles are parallelograms, but not all parallelograms are rectangles. The diagonals of a rectangle are congruent.


A rhombus is a parallelogram with four congruent sides. All rhombuses are parallelograms, but not all parallelograms are rhombuses. The diagonals of a rhombus are perpendicular to each other.


A square is a parallelogram with four right angles and four congruent sides. All squares are also parallelograms, rhombuses, and rectangles. The diagonals of a square are congruent and perpendicular to each other.


Area and Perimeter Formulas

The area of a square is found by using the formula
s is the length of one side.

The perimeter of a square is found by using the formula
, where A is the area of the rectangle, l is the length (usually considered to be the longer side), and w is the width (usually considered to be the shorter side). The numbers for l and w are interchangeable.

The perimeter of a rectangle is found by the formula
 or
l is the length and w is the width. It may be easier to add the length and width first and then double the result, as in the second formula.

The area of a parallelogram is found by the formula
 or
a and b are the lengths of the two sides.

The area of a trapezoid is found by the formula
a, b1, c, and b2 are the four sides of the trapezoid.


The area of a circle is found by the formula
r is the radius. If the diameter is given, remember to divide it in half to get the length of the radius before proceeding. The circumference of a circle is found by the formula



Any angle inscribed in a semicircle is a right angle. The intercepted arc is 180°, making the inscribed angle half that, or 90°. In the diagram below, angle ABC is inscribed in semicircle ABC, making angle ABC equal to 90°.


A chord is a line segment that has both endpoints on a circle. In the diagram below, 
 is a chord.

A secant is a line that passes through a circle and contains a chord of that circle. In the diagram below,
 is a secant and contains chord
.


A tangent is a line in the same plane as a circle, touching the circle at exactly one point. While a line segment can be tangent to a circle as part of a line that is tangent, it is improper to say a tangent can simply be a line segment that touches the circle in exactly one point. In the diagram below,
 is tangent to circle A. Notice that
 is not tangent to the circle.
 is a line segment that touches the circle at exactly one point, but if the segment were extended, it would touch the circle at a second point. The point at which a tangent touches a circle is called the point of tangency. In the diagram below, point B is the point of tangency.


A secant is a line that intersects a circle at two points.

Two secants may intersect inside the circle, on the circle, or outside the circle. When the two secants intersect on the circle, an inscribed angle is formed.

When two secants intersect inside a circle, the measure of each of two vertical angles is equal to half the sum of the two intercepted arcs. In the diagram below,
 and
.

outside a circle, the measure of the angle formed is equal to half the difference of the two arcs that lie between the two secants. In the diagram below,
.



The arc length is the length of the portion of the circumference between two points on the circle. The formula for arc length is
s is the arc length, r is the length of the radius, and
, where
 is the angular measure of the arc in radians (
).


A sector is the portion of a circle formed by two radii and their intercepted arc. While the arc length is defined exclusively as the points that are also on the circumference of the circle, the sector is the entire area bounded by the arc and the two radii.


The area of a sector of a circle is found by the formula
A is the area,
 , where
 is the measure of the central angle in degrees and r is the radius.

A circle is inscribed in a polygon if each of the sides of the polygon is tangent to the circle. A polygon is inscribed in a circle if each of the vertices of the polygon lies on the circle.

A circle is circumscribed about a polygon if each of the vertices of the polygon lies on the circle. A polygon is circumscribed about the circle if each of the sides of the polygon is tangent to the circle.

If one figure is inscribed in another, then the other figure is circumscribed about the first figure.


Solids

Area

The surface area of a solid object is the area of all sides or exterior surfaces. For objects such as prisms and pyramids, a further distinction is made between base surface area (B) and lateral surface area (LA). For a prism, the total surface area (SA) is
.


For a pyramid or cone, the total surface area is
.


The surface area of a sphere can be found by the formula
r is the radius.

Volume

The volume of a sphere is given by the formula
V=4/3πr^3
r is the radius. Both volume and surface area are generally given in terms of π.<br /> 

The volume of any prism is found by the formula
B is the area of the base and h is the height (perpendicular distance between the bases). The surface area of any prism is the sum of the areas of both bases and all sides. It can be calculated as


The volume of a rectangular prism can be found by the formula
V is the volume, l is the length, w is the width, and h is the height. The surface area of a rectangular prism can be calculated as



The volume of a cube can be found by the formula

s is the length of a side. The surface area of a cube is calculated as 6s^2
r is the radius and h is the height. The surface area of a cylinder can be found by the formula