Understanding Different Types of Numbers and Math Symbols in Algebra, Exams of Algebra

Download Understanding Different Types of Numbers and Math Symbols in Algebra and more Exams Algebra in PDF only on Docsity! Algebra I: Preliminaries and Basic Operation Complete Study Guide for Homework, Writing Papers and Taking Exams Preliminaries Before you begin learning, relearning, or reviewing algebra, you need to feel comfortable with some pre-algebra terms and operations. The first items you should become familiar with are the different categories or types of numbers and the common math symbols. Categories of numbers In doing algebra, you work with several categories of numbers.  Natural or counting numbers. The numbers 1, 2, 3, 4, …are called natural or counting numbers.  Whole numbers. The numbers 0, 1, 2, 3, …are called whole numbers.  Integers. The numbers …–2, –1, 0, 1, 2, …are called integers.  Negative integers. The numbers …–3, –2, –1 are called negative integers.  Positive integers. The natural numbers are sometimes called the positive integers.  Rational numbers. Fractions, such as   or  , are called rational numbers. Since a number such as 5 may be written as  , all integers are rational numbers. All rational numbers can be written as fractions  , with a being an integer and b being a natural number. Terminating and repeating decimals are also rational numbers, because they can be written as fractions in this form.   Irrational numbers. Another type of number is an irrational number. Irrational numbers cannot be written as fractions  , with a being an integer and b being a natural number.   and π are examples of irrational numbers. An irrational number, when exactly expressed as a decimal, neither terminates nor has a repeating decimal pattern.   Even numbers. Even numbers are integers divisible by 2: … –6, –4, –2, 0, 2, 4, 6, …   Prime numbers. A prime number is a natural number that has exactly two different factors, or that can be perfectly divided by only itself and 1. For example, 19 is a prime number because it can be perfectly divided by only 19 and 1, but 21 is not a prime number because 21 can be perfectly divided by other numbers (3 and 7). The only even prime number is 2; thereafter, any even number may be divided perfectly by 2. Zero and 1 are not prime numbers or composite numbers. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.   Odd numbers. Odd numbers are integers not divisible by 2: …–5, –3, –1, 1, 3, 5, ….   Composite numbers. A composite number is a natural number divisible by more than just 1 and itself: …4, 6, 8, 9,…   Squares. Squares are the result when numbers are multiplied by themselves, that is, raised to the second power. 2 · 2 = 4; 3 · 3 = 9. The first six squares of natural numbers are 1, 4, 9, 16, 25, 36.   Cubes. Cubes are the result when numbers are multiplied by themselves and then again by the original number, that is, raised to the third power. 2 · 2 · 2 = 8; 3 · 3 · 3 = 27. The first six cubes of natural numbers are 1, 8, 27, 64, 125, 216.  Ways to show multiplication There are several ways to show multiplication of a pair of numerical values.  When the two numerical values are known, you can show the multiplication of 4 with 3 as follows:  When one value is a number and the other value is a variable: show multiplication of 4 and a as follows:  1  2  3 7/15 What is the smallest number, greater than 1, that is both a square number and a cube number?  9  36  64 8/15 Which of the following is NOT an acceptable way to show the multiplication of x with y?  ( x)( y)  x/y  x . y 9/15 Which symbol has exactly the same meaning as “is not greater than”?  <  ≤  > 10/15 Which of the following statements is true?   11/15 Which statement says “ x is approximately equal to 5”?  x ≠ 5  x ≈ 5  x = 5 Previous 12/15 If x = 1, then which of the following statements is true?    x ≤–1 Previous 13/15 Which of the following sets does not include the number zero?  natural numbers  whole numbers  integers 14/15 The sum of two prime numbers, each greater than 10, is  always even.  always odd.  sometimes even and sometimes odd. Previous 15/15 Properties of Basic Mathematical Operations Some mathematical operations have properties that can make them easier to work with and can actually save you time. Some properties (axioms) of addition You should know the definition of each of the following properties of addition and how each can be used.  Closure is when all answers fall into the original set. If you add two even numbers, the answer is still an even number (2 + 4 = 6); therefore, the set of even numbers is closed under addition (has closure). If you add two odd numbers, the answer is not an odd number (3 + 5 = 8); therefore, the set of odd numbers is not closed under addition (no closure).   Commutative means that the order does not make any difference in the result.    Note: Commutative does not hold for subtraction.    Associative means that the grouping does not make any difference in the result.    The grouping has changed (parentheses moved), but the sides are still equal.  Note: Associative does not hold for subtraction.    The identity element for addition is 0. Any number added to 0 gives the original number.    The additive inverse is the opposite (negative) of the number. Any number plus its additive inverse equals 0 (the identity).   Some properties (axioms) of multiplication You should know the definition of each of the following properties of multiplication and how each can be used.  Closure is when all answers fall into the original set. If you multiply two even numbers, the answer is still an even number (2 × 4 = 8); therefore, the set of even numbers is closed under multiplication (has closure). If you multiply two odd numbers, the answer is an odd number (3 × 5 = 15); therefore, the set of odd numbers is closed under multiplication (has closure).   Commutative means the order does not make any difference.   a – ( b – c) = ( a – b) – c  a( b – c ) = ab – ac  ( a + b) + c = a + ( b + c) 4/15 Which of the following illustrates the identity number for addition?  5 + (–5) = 0  7 + 0 = 7  5/15 What is the only number that is its own additive inverse?  –1  0  1 6/15 What is the only number that is its own multiplicative inverse?  –1  0  1 7/15 What is the additive inverse of  ?    0 8/15 What is the multiplicative inverse of 5?  –5  1  9/15 Which one of the following sets is closed for addition?  the set of prime numbers  the set of positive odd integers  the set of positive even integers 10/15 Which of the following illustrates the associative property for multiplication?  7[(8)(9)] = [(7)(8)](9)  12(13) = 13(12)  5(2 + 3) = 5(2) + 5(3) 11/15 Which of the following is a true statement?  17 – 3 = 3 – 17  23 – (5 – 6) = (23 – 5) – 6  12[(32)(24)] = 12[(24)(32)] 12/15 If a # b = b # a for all possible values of a and b, then we would say that # has what property?  commutative property  associative property  closure property 13/15 If x is neither zero nor one, then which expression is the additive inverse of x?  – x   cannot be determined 14/15 Which of the following is a true statement?  All numbers have a multiplicative inverse.  All numbers have an additive inverse.  Both statements are true. 15/15 Multiplying and Dividing Using Zero Zero times any number equals zero. Likewise, zero divided by any nonzero number is zero. Important note: Dividing by zero is “undefined” and is not permitted.   and   are not permitted.   has no answer and   does not have a unique answer.  In neither case is the answer zero. Quiz: Multiplying and Dividing Using Zero  0  no such answer  3 1/5  3 To cube a number, just multiply it by itself twice (the exponent would be 3). For example, 5 cubed (written 53) is 5 × 5 × 5, or 125. 125 is called a perfect cube (the cube of a whole number). Following is a list of the first twelve perfect cubes.  Operations with powers and exponents To multiply two numbers with exponents, if the base numbers are the same, simply keep the base number and add the exponents.  Example 2 Multiply the following, leaving the answers with exponents. 1. 2. To divide two numbers with exponents, if the base numbers are the same, simply keep the base number and subtract the second exponent from the first, or the exponent of the denominator from the exponent of the numerator. Example 3 Divide the following, leaving the answers with exponents. 1. 2. 3. To multiply or divide numbers with exponents, if the base numbers are different, you must simplify each number with an exponent first and then perform the operation. Example 4 Simplify and perform the operation indicated. 1. 32 × 22 = 9 × 4 = 36  2. (Some shortcuts are possible.) To add or subtract numbers with exponents, whether the base numbers are the same or different, you must simplify each number with an exponent first and then perform the indicated operation.  Example 5 Simplify and perform the operation indicated. 1. 32 – 23 = 9 – 8 = 1  2. 43 + 32 = 64 + 9 = 73  If a number with an exponent is raised to another power (42)3, simply keep the original base number and multiply the exponents.  Example 6 Multiply and leave the answers with exponents. 1. 2. Quiz: Powers and Exponents 30 =  0  1  3 1/15 Which of the following has the greatest value?  5–2  (–2)0  21 2/15 Which of the following is NOT equal to  ?   123  00 3/15 Which of the following is NOT equal to  ?   123  00 3/15 Simplify 4–2  –8   5/15 32 × 33 =  35  36  95 6/15 Question7  212 Since 62 = 36 and 72 = 49, then   is between   and  . Therefore,   is a value between 6 and 7. Since 42 is about halfway between 36 and 49, you can expect that   will be close to halfway between 6 and 7, or about 6.5. To check this estimation, 6.5 × 6.5 = 42.25, or about 42. Square roots of nonperfect squares can be approximated, looked up in tables, or found by using a calculator. You may want to keep these two in mind: Simplifying square roots Sometimes you will have to simplify square roots, or write them in simplest form. In fractions,   can be reduced to  . In square roots,   can be simplified to  . There are two main methods to simplify a square root. Method 1: Factor the number under the   into two factors, one of which is the largest possible perfect square. (Perfect squares are 1, 4, 9, 16, 25, 36, 49, …) Method 2: Completely factor the number under the   into prime factors and then simplify by bringing out any factors that came in pairs. Example 2 Simplify  . In Example , the largest perfect square is easy to see, and Method 1 probably is a faster method. Example 3 Simplify  . In Example , it is not so obvious that the largest perfect square is 144, so Method 2 is probably the faster method. Many square roots cannot be simplified because they are already in simplest form, such as  ,  , and  . Do each of the following problems without the use of a calculator. Which of the following values has an exact square root?  125  49  24 1/15 Which of the following has an exact cube root?  125  49  24 2/15 Between which two integers will you find  ?  10 and 11  11 and 12  12 and 13 3/15 Between which two integers will you find  ?  4 and 5  5 and 6  6 and 7 4/15 Of the three values given, find the greatest value.    7 5/15 Of the three values given, find the least value.   11  6/15 Which row arranges the values from least to greatest?    7/15 Simplify