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Math Level 2 Fundamentals
1.1 Order of Operations The order of operations refers to the order in which you must perform the various operations in a given mathematical expression. If operations in an expression could be performed in any random order, a single expression would take on a vast array of values. As you can see, there are a great many possible evaluations of this expression depending on the order in which we perform the required operations. That’s why we have PEMDAS: a catchy acronym for determining the correct order of operations in any expression. PEMDAS stands for: - Parentheses: first, perform the operations in the innermost parentheses. A set of parentheses supersedes any other operation. - Exponents: before you do any other operation, raise all the required bases to the prescribed exponent. Exponents include square roots and cube roots, since those two operations are the equivalent of raising a base to the 1 /2 and 1/3 power, respectively. - Multiplication and Division: perform multiplication and division. - Addition and Subtraction: perform addition and subtraction. You should become familiar with some common types of numbers. Understand their properties and you will be well served. - Whole Numbers: the set of counting numbers, including zero {0, 1, 2, 3, . . .}. - Natural Numbers: the set of all whole numbers except zero {1, 2, 3, 4, 5, . . .}. - Integers: the set of all positive and negative whole numbers, including zero. Fractions and decimals are not included {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}. - Rational Numbers: the set of all numbers that can be expressed as a quotient of integers. That is, all numbers that can be expressed in the form m /n, where m and n are integers. The set of rational numbers includes all integers, and all fractions that can be created using integers in the numerator and denominator. - Irrational Numbers: the set of all numbers that cannot be expressed as a quotient of integers. Examples include: 1.01001000100001000001. . . . The sets of irrational numbers and rational numbers are mutually exclusive. Any given number must be either rational or irrational; no number can be both. - Real Numbers: every number on the number line. The set of real numbers includes all rational and irrational numbers. - Imaginary Numbers: numbers that do not appear on the real number line. Even numbers are those numbers that are divisible by two with no remainder. Odd numbers are those numbers not evenly divisible by two. Even numbers are numbers that can be written in the form 2n, where n is an integer. Odd numbers are numbers that can be written in the form 2n + 1, where n is an integer. Positive numbers are numbers that are greater than zero. Negative numbers are numbers that are less than zero. The number zero is neither positive nor negative. The absolute value of a number is the distance on a number line between that number and zero. Or, you could think of it as the positive “version” of every number. The absolute value of a positive number is that same number, and the absolute value of a negative number is the opposite of that number. The absolute value of x is symbolized by |x|. 1.2 Factors A factor is an integer that divides another integer evenly. If a/b is an integer, then b is a factor of a. To find all the factors of a number, write them down in pairs, beginning with 1 and the number you’re factoring. A prime number is a number whose only factors are 1 and itself. All prime numbers are positive (because every negative number has -1 as a factor in addition to 1 and itself). Furthermore, all prime numbers besides 2 are odd. The first few primes, in increasing order, are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, . . . To determine whether or not a number is prime, you shouldn’t check whether the number is divisible by every number less than itself. Another form of factorization is called prime factorization. The prime factorization of an integer is the listing of the prime numbers whose product is that number. To find the prime factorization of a number, divide it and all of its factors until every remaining integer is prime. This group of prime numbers is the prime factorization of the original integer. The greatest common factor (GCF) of two numbers is the greatest factor that they have in common. Finding the GCF of two numbers is especially useful in certain applications, such as manipulating fractions. The greatest common factor is the greatest integer that can be written as a product of common prime factors. That is to say, the GCF is the “overlap,” or intersection, of the two prime factorizations. Two numbers are called relatively prime if they have no common prime factors (i.e., if their GCF is 1). This doesn’t mean, however, that each number is itself prime. 8 and 15 are relatively prime because they have no common primes in their prime factorizations (8 = 2 2 2 and 15 = 3 5), but neither number is prime. 1.3 Multiples A multiple is an integer that can be evenly divided by another integer. If c/d is an integer, then c is a multiple of d. 45, 27, and 18, for example, are all multiples of 9. Alternatively, you could define a multiple as an integer with at least one factor. The least common multiple (LCM) of two integers is the smallest multiple that the two numbers have in common. Like the GCF, the least common multiple of two numbers is useful when manipulating fractions. To find the LCM of two integers, you must first find the integers’ prime factorizations. The least common multiple is the smallest prime factorization that contains every prime number in each of the two original prime factorizations. If the same prime factor appears in the prime factorizations of both integers, multiply the factor by the greatest number of times it appears in the factorization of either number. 1.4 Fractions The ability to efficiently and correctly manipulate fractions is essential. A fraction describes a part of a whole. It is composed of two expressions, a numerator and a denominator. The numerator of a fraction is the quantity above the fraction bar, and the denominator is the quantity below the fraction bar. For example, in the fraction 1 /2, 1 is the numerator and 2 is the denominator. Two fractions are equivalent if they describe equal parts of the same whole. To determine if two fractions are equivalent, multiply the denominator and numerator of one fraction so that the denominators of the two fractions are equal. As long as you multiply or divide both the numerator and denominator of a fraction by the same nonzero number, you will not change the overall value of the fraction. Fractions represent a part of a whole, so if you increase both the part and whole by the same multiple, you will not change their fundamental relationship. Reducing fractions makes life with fractions much simpler. It makes unwieldy fractions, such as 450 /600, smaller and easier to work with. To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common factor. For example, for 450/600, the GCF of 450 and 600 is 150. The fraction reduces to 3/4. In certain cases, comparing two fractions can be very simple. If they have the same denominator, then the fraction with the larger numerator is bigger. If they have the same numerator, the fraction with the smaller denominator is bigger. If the fractions do not have equal denominators, the process becomes somewhat more involved. The first step is to make the denominators the same and then to subtract as described above. The best way to do this is to find the least common denominator (LCD), which is simply the least common multiple of the two denominators. Multiplying fractions is quite simple. The product of two fractions is the product of their numerators over the product of their denominators. Multiplication and division are inverse operations. It makes sense, then, that to perform division with fractions you need to flip the second fraction over, which is also called taking its reciprocal, and then multiply the two fractions together. A mixed number is an integer followed by a fraction, like 1 and 1/2. It is another form of an improper fraction, which is a fraction greater than one. But any operation such as addition, subtraction, multiplication, or division can be performed only using the improper fraction form, so you need to know how to convert between the two. Let’s convert the mixed number 1 and 1/2 into an improper fraction. First, you multiply the integer portion of the mixed number by the denominator and add that product to the numerator. So (1 x 2) + 1 = 3, making 3 the numerator of the improper fraction. Put 3 over the original denominator, 2, and you have your converted fraction, 3 /2. 1.5 Decimals Decimals are just another way to express fractions. To produce a decimal, divide the numerator of a fraction by the denominator. As with fractions, comparing decimals can be a bit deceptive. Knowing how to convert decimals into fractions, and fractions into decimals, is a useful skill. To convert a decimal number to a fraction: 1. Remove the decimal point and use the decimal number as the numerator. 2. The denominator is the number 1 followed by as many zeros as there are decimal places in the decimal number. 3. Reduce the fraction. 1.6 Percents A percent is another way to describe a part of a whole (which means that percents are also another way to talk about fractions or decimals).Percents directly relate to decimal numbers. A percent is a decimal number with the decimal point moved two decimal places to the left. To convert from a decimal number to a percent, move the decimal point two places to the right. To convert from a fraction back to a percent, the easiest method is to convert the fraction into a decimal first and then change the resulting decimal into a percent. 1.7 Exponents An exponent defines the number of times a number is to be multiplied by itself. For example, in a^b, where a is the base and b the exponent, a is multiplied by itself b times. In a numerical example, 2^5 = 2 2 2 2 2. An exponent can also be referred to as a power: a number with an exponent of 2 is raised to the second power. The following are other terms related to exponents which you should be familiar with: - Base. The base refers to the 3 in 3^5. It is the number that is being multiplied by itself however many times specified by the exponent. - Exponent. The exponent is the 5 in 3^5. It indicates the number of times the base is to be multiplied with itself. - Square. Saying that a number is squared means that the number has been raised to the second power, i.e., that it has an exponent of 2. In the expression 6^2, 6 has been squared. - Cube. Saying that a number is cubed means that it has been raised to the third power, i.e., that it has an exponent of 3. In the expression 4^3, 4 has been cubed. In order to add or subtract numbers with exponents, you must first find the value of each power and then add the two numbers. To multiply exponential numbers raised to the same exponent, raise their product to that exponent: To divide exponential numbers raised to the same exponent, raise their quotient to that exponent: To multiply exponential numbers or terms that have the same base, add the exponents together: To divide two same-base exponential numbers or terms, just subtract the exponents: If you need to multiply or divide two exponential numbers that don’t have the same base or exponent, you’ll just have to do your work the old-fashioned way: multiply the exponential numbers out, and multiply or divide them accordingly. As we said in the negative numbers section, when you multiply a negative number by a negative number, you get a positive number, and when you multiply a negative number by a positive number, you get a negative number. These rules affect how negative numbers function in reference to exponents. - When you raise a negative number to an even number exponent, you get a positive number. For example, (-2)^4 = 16. To see why this is so, let’s break down the example. (-2)^4 means -2 -2 -2 -2. When you multiply the first two -2s together, you get 4 because you are multiplying two negative numbers. Then, when you multiply the 4 by the next -2, you get -8, since you are multiplying a positive number by a negative number. Finally, you multiply the -8 by the last -2 and get 16, since you’re once again multiplying two negative numbers. - When you raise a negative number to an odd power, you get a negative number. To see why, refer to the example above and stop the process at -8, which equals (-2)^3. There are a few special properties of certain exponents that you also should know. Zero Any base raised to the power of zero is equal to 1. If you see any exponent of the form x0, you should know that its value is 1. Note, however, that 00 is undefined. One Any base raised to the power of one is equal to itself. For example, 2¹=2,(-67)¹=-67,andx^1=x This can be helpful when you’re attempting an operation on exponential terms with the same base. Exponents can be fractions, too. When a number or term is raised to a fractional power, it is called taking the root of that number or term. This expression can be converted into a more convenient form: Seeing a negative number as a power may be a little strange the first time around. But the principle at work is simple. Any number or term raised to a negative power is equal to the reciprocal of that base raised to the opposite power. For example: 1.8 Roots and Radicals We just saw that roots express fractional exponents. But it is often easier to work with roots in a different format. When a number or term is raised to a fractional power, the expression can be converted into one involving a root in the following way: With the sign as the radical sign and xa as the radicand. Square roots are the most commonly used roots, but there are also cube roots (numbers raised to 1 /3), fourth roots, fifth roots, and so on. Each root is represented by a radical sign with the appropriate number next to it (a radical without any superscript denotes a square root). Scientific notation is a convention used to express large numbers. A number written in scientific notation has two parts: 4. A number between 1 and 10. 5. The power of 10 by which you must multiply the first number in order to obtain the large number that is being represented. The following examples express numbers in scientific notation: 1.9 Logarithms Logarithms are closely related to exponents and roots. A logarithm is the power to which you must raise a given number, called the base, to equal another number. For example, log2 8 = 3 because 2^3 = 8. In this case, 2 is the base and 3 is the logarithm. Having defined a logarithm in a sentence, let’s show it symbolically. The three equations below are equivalent: You will rarely see a test question involving basic logarithms such as log10 100 or log2 4. In particular, on the logarithm questions you’ll see in the Algebra chapter, you’ll need to be able to manipulate logarithms within equations. You should therefore know how to perform the basic operations on logarithms: The Product Rule: when logarithms of the same base are multiplied, the base remains the same, and the exponents can be added: The Power Rule: when a logarithm is raised to a power, the exponent can be brought in front and multiplied by the logarithm: You might have noticed how similar these rules are to those for exponents and roots. This similarity results from the fact that logarithms are just another way to express an exponent. The Quotient Rule: when logarithms of the same base are divided, the exponents must be subtracted: A natural logarithm is one with a base of e. The value e is a naturally occurring number, infinitely long, that can be found in growth and decay models. The natural logarithm will most likely be used in problems of growth and decay. A common numerical approximation of e is 2.718, which you could easily discover by punching e1 into your calculator. The symbol for a natural logarithm is ln, instead of log. The following three equations are equivalent: Working with natural logarithms is just like working with logarithms; the only difference is that the base for natural logarithms is always e. You might also be asked to identify the graphs of ln(x) and e(x). Take a good look at their general shapes: 2.Algebra 2.1 Distributing and Factoring Distributing and factoring are two of the most important techniques in algebra. They give you ways of manipulating expressions without changing the expression’s value. In other words, distributing and factoring are tools of reorganization. Since they don’t affect the value of the expression, you can factor or distribute one side of the equation without doing the same for the other side of the equation. The basis for both techniques is the following property, called the distributive property: Similarly, a can be any kind of term, from a variable to a constant to a combination of the two. When you “distribute” a factor into an expression within parentheses, you simply multiply each term inside the parentheses by the factor outside the parentheses. For example, consider the expression 3y(y2-6): Factoring an expression is essentially the opposite of distributing. Consider the expression 4x3-8x2+4x, for example. You can factor out the greatest common factor of the terms, which is 4x: When the product of any number of terms is zero, you know that at least one of the terms is equal to zero. For example, if xy = 0, you know that either: x = 0 and y ≠ 0, y = 0 and x ≠ 0, or x = y = 0 To solve an equation in which the variable is within absolute value brackets, you must divide the equation into two equations. The two equations are necessary because an absolute value really defines two equal values, one positive and one negative. The most basic example of this is an equation of the form |x| = c. In this case, either x=c or x = -c. A slightly more complicated example is this: |x + 3| = 5. Solve for x . In this problem, you must solve two equations: First, solve for x in the equation x + 3 = 5. In this case, x = 2. Then, solve for x in the equation x + 3 = -5. In this case, x = -8. So the solutions to the equation | x + 3| = 5 are x = {-8, 2}. Generally speaking, to solve an equation in which the variable is within absolute value brackets, first isolate the expression within the absolute value brackets, and then create two equations. Keep one of these two equations the same, while in the other equation, negate one side of the equation. In either case, the absolute value of the expression within brackets will be the same. This is why there are always two solutions to absolute value problems (unless the variable is equal to 0, which is neither positive nor negative). Before you get too comfortable with expressions and equations, we should introduce inequalities. An inequality is like an equation, but instead of relating equal quantities, it specifies exactly how two expressions are not equal. x >y - “ x is greater than y .” x <y - “ x is less than y .” x ≥ y - “ x is greater than or equal to y .” x ≤ y - “ x is less than or equal to y .” Solving inequalities is exactly like solving equations except for one very important difference: when both sides of an inequality are multiplied or divided by a negative number, the relationship between the two sides changes and so the direction of the inequality must be switched. An equation without any absolute values generally results in, at most, only a few different solutions. Solutions to inequalities are often large regions of the x - y plane, such as x <5. The introduction of the absolute value, as we’ve seen before, usually introduces two sets of solutions. The same is true when absolute values are introduced to inequalities: the solutions often come in the form of two regions of the x - y plane. Inequalities are also used to express the range of values that a variable can take on. a <x <b means that the value of x is greater than a and less than b . The rule to answer this question is the following: if either of the bounds that are being added, subtracted, or multiplied is non-inclusive (< or >), then the resulting bound is non-inclusive. Only when both bounds being added, subtracted, or multiplied are inclusive (≤ or ≥) is the resulting bound also inclusive. A range can be written by enclosing the lower and upper bounds in parentheses or brackets, depending on whether they are included in the range. Parentheses are used when the bound is not included in the range, and brackets are used when the bound is included in the range. For example, the statement a < x < b can be rewritten “the range of x is ( a , b ).” The statement a ≤ x ≤ b can be rewritten “the range of x is [ a , b ].” Finally, the statement a < x ≤ b can be rewritten “the range of x is [ a , b ].” 2.2 Systems of Equations Sometimes a question will have a lone equation containing two variables, and using the methods we’ve discussed thus far will not be enough to solve for the variables. Additional information is needed, and it must come in the form of another equation. Say, for example, that a single equation uses the two variables x and y . Try as you might, you won’t be able to solve for x or y . But given another equation with the same two variables x and y , the values of both variables can be found. Simply put, the substitution method involves finding the value of one variable in one equation and then substituting that value into the other equation to solve for the other variable. Simultaneous equations refer to equations that can be added or subtracted from each other in order to find a solution. When you solve for one variable, like we have in this last example, you can solve for the second variable using either of the original equations. If the last question had asked you to calculate the value of xy , for example, you could solve for y , as above, and then solve for x by substitution into either equation. Once you know the independent values of x and y , multiply them together. 2.3 Common Word Problems Rates Rates are ratios that relate quantities with different units. There will usually be a total quantity, an interval, and the rate of quantity/interval. Speed In speed rate problems, time (in units of seconds, minutes, or hours, etc.) is the interval, and distance (in units of inches, meters, or miles, etc.) is the total quantity. Work In work questions, you will usually find the interval measured in units of time, the total quantity measured in units of work, and the rate measured in work per time. Price In rate questions dealing with price, you will usually find the total quantity measured in a number of items, the interval measured in price (usually dollars or cents), and a rate measured in price per item. Percent Change Percent-change questions ask you to determine how a percent increase or decrease affects the values given in the question. There are two variations of percent-change questions: one where you are given the percent change and asked to find either the original value or new value; other times you will be given two of the values and asked to find the percent change. Double Percent Change A slightly trickier version of the percent-change question asks you to determine the cumulative effect of two percent changes in the same problem. Exponential Growth and Decay These types of word problems take the concept of percent change even further. For questions involving populations growing in size or the diminishing price of a car over time, you’ll need to perform a percent change over and over again. 2.4 Polynomials A polynomial is an expression that contains one or more algebraic terms, each consisting of a constant multiplied by a variable raised to a power greater than or equal to zero. For example, x 2 + 2 x + 4 is a polynomial with three terms (the third term is 4 x 0 = 4). 2 x -1, on the other hand, is not a polynomial because x is raised to a negative power. A binomial is a polynomial with exactly two terms: x + 5 and x 2 - 6 are both binomials. There is a simple acronym that is useful in helping to keep track of the terms as you multiply binomials. Evaluating a binomial expression that is raised to a power, such as ( a + b ) n , can be a dreary process when n gets to be a large number. Luckily, there is a very convenient shortcut based on Pascal’s triangle (see the next section) that will save us time and reduce the risk of error. Notice that patterns emerge when we raise the binomial ( a + b ) to consecutive powers: Looking at these patterns, we can make predictions about the expansion of ( a + b ) n . There are n + 1 terms in the expansion. For example, when the exponent, n , is 4, there are 5 terms. The power to which a is raised decreases by one each term, beginning with n and ending with 0. For example, if n = 4, then a in the second term is raised to the third power. Subsequently, the exponent of b increases by one each term, beginning with 0 and ending with n . If n = 4, then b in the second term is raised to the first power. The sum of the exponents for each term of the expansion is n . The coefficient of the n th term is equal to nCx , or the number of ways to combine n items in groups of size x , also represented as where x is the power to which either variable is raised in the n th term. Pascal’s triangle is made up of patterned rows of numbers, each row containing one more number than the last. The first row contains one number, the second row contains two numbers, the n th row contains n numbers. The first ten rows of Pascal’s triangle look like this: Each row of the triangle starts with the number one, and every interior number is the sum of the two numbers above it. Pascal’s triangle provides a very nice shortcut for dealing with the expansion of binomials: the numbers in the ( n + 1)th row of Pascal’s triangle mirror the coefficients of the terms in the expansion of ( a + b ) n . Say, for example, that you intend to expand the binomial ( f + g )5. The coefficients of the six terms in this expansion are the six numbers in the sixth row of Pascal’s triangle, which are 1, 5, 10, 10, 5, and 1: Consider the polynomials ( a + b + c ) and ( d + e + f ). To find their product, just distribute the terms of the first polynomial into the second polynomial individually, and combine like terms to formulate your final answer. A quadratic equation sets a quadratic polynomial equal to zero. That is, a quadratic equation is an equation of the form ax 2 + bx + c = 0. The values of x for which the equation holds are called the roots, or solutions, of the quadratic equation. Most of the questions on quadratic equations involve finding their roots. To factor a quadratic you must express it as the product of two binomials. They are the perfect square and the difference of two squares. If you memorize the formulas below, you may be able to avoid the time taken by factoring. There are two kinds of perfect square quadratics. They are: a 2 + 2 ab + b 2 = ( a + b )( a + b ) = ( a + b )2. Example: a 2 + 6 ab + 9 = ( a + 3)2 a 2 - 2 ab + b 2 = ( a - b )( a - b ) = ( a - b )2.. Example: a 2 - 6 ab + 9 = ( a - 3)2 Note that when you solve for the roots of a perfect square quadratic equation, the solution for the equation ( a + b )2 = 0 will be - b , while the solution for ( a - b )2 = 0 will be b . Factoring using the reverse-FOIL method is really only practical when the roots are integers. Quadratics, however, can have decimal numbers or fractions as roots. Equations like these can be solved using the quadratic formula. For an equation of the form ax 2 + bx + c = 0, the quadratic formula states: If you want to find out quickly how many roots an equation has without calculating the entire formula, all you need to find is an equation’s discriminant. The discriminant of a quadratic is the quantity b 2 - 4 ac . As you can see, this is the radicand in the quadratic equation. If: b 2 - 4 ac = 0, the quadratic has one real root and is a perfect square. b 2 - 4 ac >0, the quadratic has two real roots. b 2 - 4 ac <0, the quadratic has no real roots and two complex roots. 3.Plane Geometry 3.1 Lines and Angle -line segment -midpoint -equidistant angle -Supplementary angles, complementary angles -Alternate exterior angles, alternate interior angles -Corresponding angles -perpendicular lines -parallel lines 3.2 triangles -scalene triangle, isosceles triangle, equilateral triangle, right triangle 3.3 Polygons -quadrilateral, trapezoid -parallelogram -rectangle 3.4Circles -radius, diameter, chord -central angle -circumference 4. coordinate plane 4.1 The Coordinate Plane 4.2 Lines and Distance 4.4 Lines 4.5 Other Important Graphs and Equations 5.Trigonometry 5.1Basic Functions -Sine, cosine, tangent 5.2 Solving Right Triangles 5.3 Trigonometric Identities 5.4 The Unit Circle 5.5 Radians and Degrees 5.6Graphing in the Entire Coordinate Plane 6. Functions 6.1Characteristics of a Function 6.2 Operations on Functions 6.3 Domain and Range 6.4 Graphing Functions -asymptote -hole -roots (or zeroes) -Odd Functions -Symmetry Across the x-axis 7. Statistics 7.1 Statistical Analysis arithmetic, mean, median, mode 7.2 Probability -probability 7.3Permutations and Combinations -factorial, permutation, combination 7.4 Sets -union -intersection |