MATH030 M1 Lesson 1 The Real Number System.pdf

COLLEGE ALGEBRA MATH030 Excellence and Relevance  THE REAL NUMBER SYSTEM M1 LESSON 1 Excellence and Relevance  SET A set is a group or col...

COLLEGE ALGEBRA MATH030 Excellence and Relevance  THE REAL NUMBER SYSTEM M1 LESSON 1 Excellence and Relevance  SET A set is a group or collection of objects that are called members or elements of the set. Ex. The starting lineup on a baseball team is a subset of the entire team. The set of natural numbers, {1, 2, 3, 4, If every member of set B is also a …}, is a subset of the set of whole member of set A, then we say B is numbers, {0, 1, 2, 3, 4, …}, which is a subset of the set of integers, {…, −4, a subset of A and denote it as −3, −2, −1, 0, 1, 2, 3, …}. B ⊂ A. Excellence and Relevance  SET If a set has no elements, it is called the empty set, or null set, and is denoted by the symbol: The set of real numbers consists of two main subsets: Ø Rational & Irrational numbers Excellence and Relevance  THE REAL NUMBER SYSTEM REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS TRANSCENDENTAL INTEGERS FRACTION NUMBERS SURD ! (…-2, -1, 0, 1, 2, …) (2/3, 5/4, 12/17) 𝟏𝟏 𝟐 𝟕 𝝅 𝒆 WHOLE NUMBERS NEGATIVE NUMBERS (0, 1, 2, 3, 4, 5 …) (-1, -2, -3, -4, -5 …) POSITIVE INTEGERS/ ZERO NATURAL NUMBERS (0) Excellence and Relevance (1, 2, 3, 4, 5 …)  THE REAL NUMBER SYSTEM SYMBOL NAME DESCRIPTION EXAMPLES N Natural Numbers Counting Numbers 1, 2, 3, 4, 5, … W Whole Numbers Natural Numbers and Zero 0, 1, 2, 3, 4, 5, … Whole Numbers and Negative Natural …, −3, −2, −1, 0, 1, 2, 3, … Z Integers Numbers Ratios of Integers: −19 1 ! (𝑏 ≠ 0) −17, − , 0, , 1.37, " 7 3 Q Rational 3.666/ Numbers Ø Decimal representation terminates, or Ø Decimal representation repeats # I Irrational Numbers whose decimal representation 2, 2, 1.2179 … , 𝜋, 𝑒 Numbers does not terminate or repeat R Real Numbers Rational and Irrational Numbers 2 𝜋, 5, − , 17.25, 7 3 Excellence and Relevance  COMPLEX NUMBERS Ø combination of a real number and an imaginary number COMPLEX NUMBERS 𝟐 − 𝟑𝒊, 𝟐𝒊, 𝟏𝟔, 𝝅 RECTANGULAR FORM POLAR FORM 𝒙 + 𝒚𝒊 or 𝒙 + 𝒋𝒚 𝒓∠𝜽 TRIGONOMETRIC FORM EXPONENTIAL FORM 𝒓 cos 𝜽 + 𝒋 sin 𝜽 𝒓𝒆𝒋𝜽 𝒓𝒄𝒋𝒔𝜽 Excellence and Relevance  THE REAL NUMBER SYSTEM Please take note that: ØEvery real number is either a rational number or an irrational number. ØThe real number system consists of the positive numbers, the negative numbers, and zero. Excellence and Relevance  RATIONAL AND IRRATIONAL NUMBERS Note that: ! Rational Numbers include all integers or all fractions that are ratios of integers. For example, " and 3. 6# are rational numbers, while 2 and 3.2179 … are irrational. The ellipsis following the last decimal digit denotes continuing in an irregular fashion, whereas the absence of such dots to the right of the last decimal digit implies that the decimal expansion terminates. Example: RATIONAL CALCULATOR DECIMAL DESCRIPTION NUMBER DISPLAY REPRESENTATION (FRACTION) ! 3.5 3.5 Terminates " #$ 1.25 1.25 Terminates #" " 0.666666666 0. 6/ Repeats % # 0.09090909 Repeats 0.09 ## Excellence and Relevance  REAL NUMBER LINE −4 −3 −2 −1 0 1 2 3 4 The real number line is a graph used to represent the set of all real numbers. Excellence and Relevance  SAMPLE PROBLEMS a. Is it true that all integers are rational numbers? b. Let’s classify the following real numbers as rational or irrational: 1 1 8 ! E −3, 0, , 3, 𝜋, 7.51, , − , 6.66666, 2, 12, 0, 2.010010001 … , 𝑒 4 3 5 Excellence and Relevance  SOLUTION a. Yes! It is true that all integers are rational numbers. b. Rational: ' ' * −3, 0, , , 7.51, − , 6.66666, , 12, 0 ( ) + Irrational: ! 3, 𝜋, 2, 2.010010001 … , 𝑒 Excellence and Relevance  PROPERTIES OF REAL NUMBERS Basic Rules of Algebra NAME DESCRIPTION MATH (Let a, b, and c EXAMPLE each be any real number) Commutative property of Two real numbers can be 𝑎+𝑏 =𝑏+𝑎 3𝑥 + 5 = 5 + 3𝑥 addition added in any order. Commutative property of Two real numbers can be 𝑎𝑏 = 𝑏𝑎 𝑦 8 3 = 3𝑦 multiplication multiplied in any order. Associative property of When three real numbers 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐) 𝑥 + 5 + 7 = 𝑥 + (5 + 7) addition are added, it does not matter which two numbers are added first. Excellence and Relevance  PROPERTIES OF REAL NUMBERS Basic Rules of Algebra NAME DESCRIPTION MATH (Let a, b, and c EXAMPLE each be any real number) Associative property of When three real numbers 𝑎𝑏 𝑐 = 𝑎(𝑏𝑐) −3𝑥 𝑦 = −3(𝑥𝑦) multiplication are multiplied, it does not matter which two numbers are multiplied first. Distributive property Multiplication is 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 5 𝑥 + 2 = 5𝑥 + 10 distributed over all the 𝑎 𝑏 − 𝑐 = 𝑎𝑏 − 𝑎𝑐 5 𝑥 − 2 = 5𝑥 − 10 terms of the sums or differences within the parentheses. Excellence and Relevance  PROPERTIES OF REAL NUMBERS Basic Rules of Algebra NAME DESCRIPTION MATH (Let a, b, and c EXAMPLE each be any real number) Additive identity property Adding zero to any real 𝑎+0=𝑎 7𝑦 + 0 = 7𝑦 number yields the same 0+𝑎 =𝑎 real number. Multiplicative identity Multiplying any real 𝑎81=𝑎 8𝑥 1 = 8𝑥 property number by 1 yields the 18𝑎 =𝑎 same real number. Excellence and Relevance  PROPERTIES OF REAL NUMBERS Basic Rules of Algebra NAME DESCRIPTION MATH (Let a, b, and c EXAMPLE each be any real number) Additive inverse property The sum of a real number 𝑎 + (−𝑎) = 0 4𝑥 + −4𝑥 = 0 and its additive inverse (opposite) is zero. Multiplicative inverse property The product of a nonzero 1 1 𝑎8 =1𝑎 ≠0 𝑥+2 8 =1 real number and its 𝑎 𝑥+2 multiplicative inverse where: (reciprocal) is 1. 𝑥 ≠ −2 Excellence and Relevance  PROPERTIES OF NEGATIVES DESCRIPTION MATH (Let a, b, and c each be any EXAMPLE real number) A negative quantity times a positive −𝑎 𝑏 = −𝑎𝑏 −8 3 = −24 quantity is a negative quantity. A negative quantity divided by a positive −𝑎 𝑎 −16 =− = −4 quantity is a negative quantity. 𝑏 𝑏 4 or or or 𝑎 𝑎 15 A positive quantity divided by a negative =− = −5 quantity is a negative quantity. −𝑏 𝑏 −3 A negative quantity times a negative −𝑎 −𝑏 = 𝑎𝑏 −2𝑥 −5 = 10𝑥 quantity is a positive quantity. A negative quantity divided by a −𝑎 𝑎 −12 = =4 negative quantity is a positive quantity. −𝑏 𝑏 −3 The opposite of a negative quantity is a − −𝑎 = 𝑎 − −9 = 9 positive quantity (subtracting a negative quantity is equivalent to adding a positive quantity). A negative sign preceding an expression − 𝑎 + 𝑏 = −𝑎 − 𝑏 −3 𝑥 + 5 = −3𝑥 − 15 is distributed throughout the expression. − 𝑎 − 𝑏 = −𝑎 + 𝑏 −3 𝑥 − 5 = −3𝑥 + 15 Excellence and Relevance  SAMPLE PROBLEMS Simplify the following expressions using the distributive property: 1. −2𝑎 5 − 6𝑏 𝑨𝒏𝒔: −𝟏𝟎𝒂 + 𝟏𝟐𝒂𝒃 2. 5𝑥 − 2𝑦 − 3𝑧 + (3𝑧 − 2𝑦) 𝑨𝒏𝒔: 𝟓𝒙 − 𝟒𝒚 + 𝟔𝒛 ! 𝟏 3. 𝑚 + 4 3𝑛 + !" − 3𝑝 𝑨𝒏𝒔: 𝒎 + 𝟏𝟐𝒏 + 𝟒 − 𝟏𝟐𝒑 Excellence and Relevance  LAWS OF EXPONENTS Product Rule 𝑎% ×𝑎& = 𝑎%'& Quotient Rule 𝑎% ÷ 𝑎& = 𝑎%(& Power of a Power Rule (𝑎% )& = 𝑎%& Power of a Product Rule (𝑎𝑏)% = (𝑎% 𝑏 % ) ) % )" Power of a Quotient Rule ( ) = * *" + Zero Exponent Rule 𝑎 =1 ! Negative Exponent Rule 𝑎(% = )" " # Fractional Exponent Rule 𝑎 = # 𝑎% Excellence and Relevance  SAMPLE PROBLEMS Simplify the following: , ,-$.! 𝒙𝟒 1. 𝑨𝒏𝒔: "-!. 𝟗𝒂𝟐 , 2 𝟔 𝟑 2. (𝑎 𝑥) 𝑨𝒏𝒔: 𝒂 𝒙 𝟗𝒙𝟔𝒕 3. 3𝑥 25(! , 𝑨𝒏𝒔: 𝒙𝟐 (6 ) 97 + 8 ! (𝟐𝒙𝟑 𝒚 4. ,7 * 8 + 6* 𝑨𝒏𝒔: 𝒛𝟑 6 $#-. 7 /#0! 5. 6 $#0!7 $#0* 𝑨𝒏𝒔: [𝒙𝟐 𝒚(𝒏'𝟏) ]𝟐 Excellence and Relevance  END OF LESSON Excellence and Relevance CREDITS TO: ENGR. FROILAN N. JIMENO II, ECE, ECT