Math Review (All Important Accuplacer Math Things To Know)

This is all the basic mathematical concepts that you need to k ow for the Math portion on the Accuplacer test. 

Properties of Real Numbers

Closure
• a + b is a real number; when you add two real numbers, the result is also a real number
Example: 3 and 7 are both real numbers, 3+7=10 and the sum, 10, is also a real number.
• a – b is a real number; when you subtract two real numbers the result is also a real number.
Example: 2 and 5 are both real numbers, 2 – 5 = -3, and the difference, -3, is also a real number.
• (a)(b) is a real number; when you multiply two real numbers, the result is also a real number.
Example: 9 and -2 are both real numbers; (9)(-2) = -18, and the product , -18 is also a real number.
• a / b is a real number when b ≠ 0; when you divide two real numbers, the result is also a real number unless the denominator (divisor) is zero.
Example: -10 and 5 are both real numbers, -10 / 5 = -2, and the quotient, -2, is also a real number.

Commutative
• a + b = b + a; you can add numbers in either order and get the same answer.
Example: 2 + 6 = 8 and 6 + 2 = 8 so 2 + 6 = 6 + 2
• (a)(b) = (b)(a); you can multiply numbers in either order and get the same answer. Example: (7)(10) = 70 and (10)(7) = 70 so (7)(10) = (10)(7)
• a – b ≠ b – a; you cannot subtract in any order and get the same answer.
Example: 4 – 6 = -2, but 6 – 4 = 2. There is no commutative property for subtraction.
• a / b ≠ b/a; you cannot divide in any order and get the same answer. Example :
4/2 = 2, but 2/4 =.5 so there is no commutative property for division.

Associative
• (a + b) + c = a + (b + c); you can group any numbers in any arrangement when adding and get the same answer. Example: (1 + 2) + 3 = 3 + 3 = 6 and 1 + (2 + 3)
= 1 + 5 = 6 so (1 + 2) + 3 = 1 + (2 + 3).
• (ab)c = a(bc); you can group any numbers in any arrangement when multiplying and get the same answer. Example: (2 x 6)3 = (12)3 = 36 and 2(6 x 3) = 2(18) = 36 so (2 x 6)3 = 2(6 x 3)
• The associative property does not work for subtraction or division.

Identities 
• a + 0 = a; zero is the identity for addition because adding zero does not change the original number.
Example: 7 + 0 = 7 and 0+7 = 7.
• a(1) = a; one is the identity for multiplication because multiplying by one does not change the original number.
Example:
21(1) = 21 and(1)21=21.
• Identities for subtraction and division become a problem. It is true that 29 – 0 = 29, but 0 – 29 = -29, not 29. This is also the case for division because 4/1 = 4, but 1/4 = .25, so the identities do not hold when the numbers are reversed.

Inverses
• a + (-a) = 0; a number plus its additive inverse (the numbers with the opposite sign) will always equal zero. Example: 6 + (-6) = 0 and (-6) + 6 = 0. The exception is zero because 0 + 0 = 0 already.
• a(1/a) = 1; a number time its multiplicative inverse or reciprocal ( the number written as a fraction and flipped) will always equal one. Example: 5(1 / 5) = 1.
The exception is zero because zero cannot be multiplied by any number and result in a product of one/

Distributive Property
• a(b + c) = ab + ac or a(b - c) = ab - ac; each term in the parentheses must be multiplied by the term in front of the parentheses. Example: 4(5 + 7) = 4(5) +4(7) = 20 + 28 = 48. This is a simple example and the distributive property is not required to find the answer. When the problem involves a variable however, the distributive property is a necessity. Example: 4(5a + 7) = 4(5a) + 4(7) = 20a +28.

Properties of Equality 
• REFLEXIVE : a = a; both sides of the equation are identical EXAMPLE:6+k = 6+k
• SYMMETRIC: If a = b then b = a. This property allows you to exchange the two sides of an equation. Example: 4a – 7 = 9 - 7a + l5 becomes 9 - 7a + 15 =4a - 7.
• TRANSITIVE; If a = b and b = c the a = c. This property allows you to connect statements which are each equal to the same common statement.
Example: 5a - 6 = 9k and 9k = a + 2; you can eliminate the common term 9k and connect the following into one equation: 5a - 6 = a+2.
• ADDITION PROPERTY OF EQUALITY: If a = b then a + c = b + c. This property allows you to add any number or algebraic term to any equation as long as you add it to both sides to keep the equation true. EXAMPLE: 5 = 5; if you add 3 to one side and not the other the equation becomes 8 = 5 which is false, but if you add 3 to both sides you gets true equation 8 = 8. Also, 6a + 2 = 14 becomes
6a + 2 + (-2) = 14 + (-2) if you add 2 to both sides. The results in the equation 6a
= 12

Multiplcation Property of Equality:

If a = b then ac = bc when c ≠ 0. This property allows you to multiply both sides of an equation by any nonzero value. EXAMPLE: If 4a = -24, then (4a)(.25 ) = (-24)(.25) and a = -6. Notice that both sides of the = were multiplied by 25.

Definitions

• NATURAL or Counting NUMBERS: {1 2,3,4,5,…, ll, l2,...}
• WHOLENUMBERS: {0, 1, 2 ,3,…, 10, 11, 12, 13, ...}
• INTEGERS:{....-4,-3,-2,-l,0,l,2,3,4,...}
• RATIONAL NUMBERS: {p/q | p and q are integers, q ≠ 0}; the sets of Natural numbers, Whole numbers, and Integers, as well as numbers which can be written as proper or improper fractions, are all subsets of the set of Rational Numbers.
• IRRATIONAL NUMBERS: {x | x is a real number but is not a Rational number }; the sets of Rational numbers and Irrational numbers have no elements in common and are therefore disjoint sets.
• REAL NUMBERS: {x | x is the coordinate of a point on a number line}; the union of the set of Rational numbers with the set of Irrational numbers equals the set of Real Numbers.
• IMAGINARY NUMBERS: {ai | a is a real number and i is the number whose square is -1 }; i² = -1: the sets of Real numbers and Imaginary numbers have no elements in common and are therefore disjoint sets.
• COMPLEX NUMBERS: {a + bi | a and b are real numbers and i is the number whose square is -1 }; the set of Real numbers and the set of Imaginary numbers are both subsets of the set of Complex numbers. EXAMPLES: 4 + 7i and 3 - 2i are complex numbers.

Operations of real Numbers
VOCABULARY
• TOTAL or SUM is the answer to an addition problem. The numbers added are called addends EXAMPLE: In 5 + 9=14, 5 and 9 are addenda and 14 is the total or sum.
• DIFFERENCE is the answer to a subtraction problem. The number subtracted is called the subtrahend. The number from which the subtrahend is subtracted is called the minuend. EXAMPLE: In 25 - 8 = 17, 25 is the minuend, 8 is the subtrahend, and 17 is the difference.
• PRODUCT is the answer to a multiplication problem. The numbers multiplied are each called a factor. EXAMPLE: In 15 x 6 = 90, 15 and 6 are factors and 90 is product.
• QUOTIENT is the answer to a division problem. The number being divided is called the dividend. The number that you are dividing by is called the divisor. If there is a number remaining after the division process has been completed, that number is called the remainder. Example: In 45 ÷ 5 = 9 , which may also be written as 45/5,45 is the dividend, 5 is the divisor and 9 is the quotient.

An EXPONENT indicates the number of times the base is multiplied by itself; that is, used as a factor. EXAMPLE: In 5², 5 is the base and 2 is the exponent, or power, and 5² = (5)(5) = 25, notice that the base, 5, was multiplied by itself 2 times.

PRIME NUMBERS are natural numbers greater than 1 having exactly two factors, itself and one. EXAMPLES: 7 is prime because the only two natural numbers that multiply to equal 7 are 7 and 1; 13 is prime because the only two natural numbers that multiply to equal 13 are 13 and 1.

COMPOSITE NUMBERS are natural numbers that have more than two factors.

Examples: 15 is a composite number because 1, 3, 5, and 15 all multiply in some combination to equal 15; 9 is composite because 1. 3, and 9 all multiply in some combination to equal 9. The GREATEST COMMON FACTOR (GCF)or greatest common divisor (GCD) of a set of numbers is the largest natural number that is a factor of each of the numbers in the set; that is, the largest natural number that will divide into all of the numbers in the set without leaving a remainder. EXAMPLE: The greatest common factor (GCF) of 12,30 and 42 is
6 because 6 divides evenly into 12,30, and 42 without leaving remainders.

The LEAST COMMON MULTIPLE (LCM) of a set of numbers is the smallest natural number that can be divided (without remainders) by each of the numbers in the set. Example: The least common multiple of2, 3, and4 is 12 because although 2, 3, and 4 divide evenly into many numbers including 48, 36, 24, and
12, the smallest is 12.

The DENOMINATOR of a fraction is the number in the bottom; that is.the divisor of the indicated division of the fraction. EXAMPLE.- In 5/8, 8 is the denominator and also the divisor in the indicated division.

The NUMERATOR of a fraction is the number in the top; that is, the dividend of indicated division of the fraction. EXAMPLE: In 3/4, 3 is the numerator and also the dividend in the indicated division.

ORDER OF OPERATIONS
DESCRIPTION: The order in which addition, subtraction, multiplication, and division are performed determines the answer.

ORDER

1. Parentheses: Any operations contained in parentheses are done first, if there are any. This also applies to these enclosure symbols { } and [].

2. Exponents: Exponent expressions are simplified second, if there are any.

3. Multiplication and Division: These operations are done next in the order in which they are found, going left to right; that is, if division comes first going left to right, then it is done first.

4. Addition and Subtraction: These operations are done next in the order in which they are found going left to right; that is, if subtraction comes first, going left to tight, then it is done first.

DECIMAL NUMBERS

The PLACE VALUE of each digit in a base ten number is determined by its position with respect to the decimal point. Each position represents multiplication by a power of ten. Example: In 324, 3 means 300 because it is 3 times 10² (10² = 100). 2 means 20 because it is 2 times 10 (10 = 10). And 4 means 4 times one because it is 4 times l0º(l0º = 1). There is an invisible decimal point to the right of the 4. In 5.82, 5 means 5 times one because it is 5 times l0º (l0º = 1), 8 means 8 times one tenth because it is 8 times 10-1 (10-1 = .1 = 1/10), and 2 means 2 times one hundredth because it is 2 times l0-²~ (l0-² =.01 = 1/100).

PLACE VALUE
In the number 3025.6789
3 is the thousands place
0 is the hundreds place
2 is the tens place
5 is the ones or units
.6 is the tenths place
.67 is the hundredths place
.678 is the thousandths place
.6789 is the ten-thousandths place

WRITING DECIMAL NUMBERS AS FRACTIONS
Write the digits that are behind the decimal point as the numerator (top) of the fraction
Write the place value of the last digit as the denominator (bottom) of the fraction.
Any digits in front of the decimal point are whole numbers
Example: In 4.068, the last digit behind the decimal point is 8 and it is in the 1000ths place; therefore, 4.068 becomes 4 68/1000
Notice the number of zeros in the denominator is equal to the number of digits behind the decimal point in the original number

ADDITION
• Write the decimal numbers in a vertical form with the decimal points lined up one under the other, so digits of equal place value are under each other.

SUBTRACTION
• Write the decimal numbers in a vertical form with the decimal points lined up one under the other.
• Write additional zeros after the last digit behind the decimal point in the minuend (number on top) if needed (both the minuend and the subtrahend should have an equal number of digits behind the decimal point.)

MULTIPLICATION
• Multiply
• Count the number of digits behind the decimal point in all factors.
• Count the number digits behind the decimal point in the answer. The answer must have the same number of digits behind the decimal point as there are digits behind the decimal points in all the factors. It is not necessary to line the decimal points up in multiplication.

RATIO, PROPORTION, & PERCENT

Ratio
• Definition: Comparison between two quantities
• Forms: 3 to 6, 3 : 6, 3/6

Proportion
• Definition: Statement of equality between two ratios of fractions.
• Forms: 3 : 6 :: 6 : 12, 3/6 = 6/12

Percents
• Definition: Percent means “out of 100” or “per 100”
• Percent and equivalent fractions

1. Percents can be written as fractions by placing the number over 100 and simplifying or reducing. Example: 20% = 20/100 = 2/10 ; 4.5% =4.5/100
= 45/1000 = 9/200

2. Fractions can be changed to percents by writing them with denominators of 100. The number is then the percent number. Example: 3/5 = 3/5 x
20/20 = 60/100 = 60%
• Percents and decimal numbers

1. To change a percent to a decimal number; move the decimal point two places to the left because percent means “out of 100” and decimal numbers with two digits behind the decimal point also mean “out of 100”.

2. To change a decimal number to a percent move the decimal point two places to the right

FRACTIONS
Reducing
• Divide numerator (top) and denominator (bottom) by the same number, thereby renaming it to an equivalent fraction in lower terms This process may be repeated

ADDITION a/c + b/c = a + b/c where c ≠ 0

Change to equivalent fractions with common denominator.

1. 1. Find the least common denominator by determining the smallest number which can be divided evenly (no remainders)by all of the numbers in the denominators (bottoms).

2. Multiply the numerator and denominator of each fraction so the fraction value has not changed but the common denominator has been obtained.

3. Add the numerators and keep the same denominator because the addition of fractions is counting equal parts.

Subtraction a/c - b/c = a - b/c where c ≠ 0
Change to equivalent fractions with a common denominator.

1. Find the least common denominator by determining the smallest number which can be divided evenly by all of the numbers in the denominators (bottoms).

2. Multiply the numerator by the same number so the fraction value has not changed, but the common denominator has been obtained

3. Subtract the numerators and keep the same denominator because subtraction of fractions is finding the difference between equal parts.

Multiplication a/c * b/d = a * b / c* d where c ≠ 0 and d ≠ 0
• Common denominators are NOT needed.

1. I .Multiply the numerators (tops) and multiply the denominators (bottoms) then reduce the answer to lowest terms.

2. OR - reduce any numerator (top) with any denominator (bottom) and then multiply the numerators and multiply the denominators.

DIVISION (a/c) /(b/d) = (a/c) x (d/b) = (a * d) / (c * b) where b≠ 0, c ≠ 0 and d ≠ 0
• Common denominators are NOT needed.

1. Change division to multiplication by the reciprocal; that is, flip the fraction in back of the division sign and change the division sign to a multiplication sign.

2. Now follow the steps for multiplication of fractions as indicated above.

Algebra Vocabulary

• Variables are letters used to represent numbers.
• Constants are specific numbers that are not multiplied by any variables.
• Coefficients are numbers that are multiplied by one or more variables.
• Term. are constants or variable expressions.
• Like or similar terms are terms that have the same variables to the same degree or exponent value. Coefficients do not matter, they may be equal or not.
• Algebraic expressions are terms that are connected by either addition or subtraction.
• Algebraic equations are statements of equality between at least two terms.