REAL NUMBERS ASSOCIATIVE o o o o Know It NO~;~ It Always' o • operations of real numbers • fractions • mixed numbers • ratio • proportion • algebra • and much more o o o em Square Area: A = hb If h=8 then b=8 also, as all sides are equal in a square, then: A=64 square units Rectangular Prism Volwne ~ h w V=lwh; Ifl=12 w=3, h=4 then: I V=(12)(3)(4) V= I44 cubic units Rectangle Area Ifh=4 and b= 12 then: A = (4)(12) A=48 square units Cube Volume, V=e l Each edge length, e, is equal to the other e edges in a cube If e=8 then: V=(8)(8)(8), V=512 cubic units V=rrr 2h TriangleArea:A= I/2 b~ Cylinder Volume , If radius, r=9, h=8 then: lfh=8 and b=12 then: • V=rr(9)2(8), V=3 14(81 )(8), A= 112 (8)(12) : V=2034.72 cubic units A=48 square units b Cone Volume, V= I /3~h Parallelogram Area: A=hb; If r=6 and h=8 then: Ifh=6 and b=9 h / V= I/3rr(6)2(8), then: A=(6)(9) b V= 1/3(3.14)(36)(8) A=54 square umts V=30 1.44 cubic units ;16 o o a + b = b + a; you can add numbers in either order and get the same answer EXAMPLE: + 15 = 24 and 15 + = 24 so + 15 = 15 + (a)(b) = (b)(a); you can multiply numbers in either order and get the same answer EXAMPLE: (4)(26) = 104 and (26)(4) = 104 so (4)(26) = (26)(4) a - b b - a; you cannot subtract in any order and get the same answer EXAMPLE: - = 6, but - = -6 There is no commutative property for subtraction alb b/a; you cannot divide in any order and get the same answer EXAMPLE : 8/2 = , but 2/8 = 25 so there is no commutative property for division W e h • o a + (-a) = 0; a number plus its additive inverse (the number with the opposite sign) will always equal zero EXAMPLE: + (-5) = and (-5) + = O The exception is zero because + = already a (1/a) = I ; a number times its multiplicative inverse or reciprocal (the number written as a fraction and flipped) will always equal one EXAMPLE: 5(1 /5) = I The exceptIOn is zero because zero cannot be multiplied by any number and result in a product of one o o o a(b + c) = ab + ac or alb - 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 in order 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 IDENTITIES o o a + = a; zero is the identity for addition because adding zero does not change the original number EXAMPLE: + = and + = a (I) = a; one is the identity for multiplication because multiplying by one does not change the onginal number EXAMPLE: 23 (I) = 23 and (Il 23 = 23 Identities for subtraction and division become a problem It is true that 45 - =45, but - 45 = -45 not 45 This is also the case for division because 411 = , but 1/4 = 25 so the identities not hold when the numbers are reversed REFLEXIVE : a = a; both sides of the equation are identical EXAMPLE: + k = + k SYMMETRIC : If a = b then b = a This property allows you to exchange the two sides of an equation EXAMPLE: 4a - = - 7a+15 becomes - 7a + 15 = 4a - TRANSITIVE: If a = band b = e then a = e This property allows you to connect statements which are each equal to the same common statement EXAMPLE: 5a - = 9k and 9k = a + then you can eliminate the common term 9k and connect the following into one equation: 5a - = a + ADDITION PROPERTY OF EQUALITY: If a = b then a + c = b + e 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: = and if you add to one side and not the other the equation becomes = which is false, but if you add to both sides you get a true equation = Also, 5a + = 14 becomes 5a + + (-4) = 14 +(-4) if you add -4 to both sides This results in the equation 5a = 10 MULTIPLICATION PROPERTY OF EQUALITY: If a = b then ac = be when e O 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 then a = -6 Notice that both sides of the = were multiplied by 25 OPERATIONS OF INTEGERS ABSOLUTE VALUE o o (a + b) + e = a + (b + c); you can group numbers in any arrangement when adding and get the same answer EXAMPLE: (2 + 5) + = + 9= 16 and + (5 + 9) = + 14 = 16 so (2 + 5) + = + (5 + 9) (ab)e = a(be); you can group numbers in any arrangement when multiplying and get the same answer EXAMPLE: (4x5)8=(20)8=160 and4(5x8)=4(40)= I60 so (4x5)8 = 4(5x8) The associative property does not work for subtraction or division EXAMPLES: (10 - 4) - 2= - = 4, but 10 - (4 - 2) = 10 - = for division (1 2/6)/2 = (2)/2 = I, but 121(6/2) = 12/3=4 Notice that these answers are not the same _I ',1 o PERIMETER: The perimeter, P, of a two-dimensional shape is the sum of all side lengths AREA: The area, A, of a two·dimensional shape is the number of square units that can be put in the re~ion enclosed by the sides NOTE: Area is obtained through some combination of multiplying heights and bases , which always form 90° angles with each other, except in circles VOLUME: The volume, V of a three·dimensional shape is the number cubic units that can be put in the region enclosed by all the sides e o o COMMUTATIVE o • geometric formulas • properties of real numbers • operations of integers -addition -double negative -subtraction -multiplication and division a + b is a real nwnber, when >00 add real nwnbers the result is also a real nwnber EXAMPLE: and are both real numbers, + = and the sum, 8, is also a real number a - b is a real number; when you subtract real numbers the result is also a real number EXAMPLE: and II are· both real numbers, - II = -7, and the difference, -7, is also a real number (a)(b) is a real number; when you multiply real numbers the result is also a real number EXAMPLE: 10 and -3 are both real numbers, (10)(-3) = -30, and the product, -30, is also a real number a I b is a real number when b 0; when you divide real numbers the result is also a real number unless the denominator (divisor) is zero EXAMPLE: -20 and are both real numbers, -20/5 = - 4, and the quotient, - 4, is also a real number Definition: 1x 1= x i(x > or x = and 1x 1= -x if x < 0; that is, the absolute value ofa number is always the positive value of that number EXAMPLES: 161 = and 1-61= 6, the answer is a positive in both cases ''I'] • If the signs of the numbers are the same, ADD The answer has the same sign as the numbers EXAMPLES: (-4) + (-9) = -13 and + II = 16 o If the signs of the numbers are different, SUBTRACT The answer has the sign of the larger number (ignoring the signs or taking the absolute value of the numbers to determine the larger number) EXAMPLES: (-4)+(9) =5 and (4)+(-9) = -5 o DOUBLE NEGATIVE o -(-a) = a that is, the sign in front of the parentheses changes the sign of the contents of the parentheses EXAMPLES: -(-3) = +3 or -(3) = -3; also, -(5a - 6) = -5a + OPERATIONS OF INTEGERS CONTINUED: SUBTRACTION 11" i' 'PY'1t i (.1' j;! it M"]C" it i j;! Y' jilt OPERATIONS OF REAL NUMBERS 1] The Fundamental Theorem of Arithmetic states that every composite number can be expJ\!ssed as a unique product of prime numbers EXAMPLES: 15 = (3)(5) where 15 is composite and both and are prime;72 = (2)(2)(2)(3)(3) where 72 is ·composite and both and are prime notice that 72 also equals (8)(9) but this docs not demonstrate the theorem because neither nor are prime numbers VOCABULARY • Change subtraction to addition ofthe opposite number; a - b = a + (-b); that is, change the subtraction sign to addition and also change the sign of the number directly behind the subtraction sign to the opposite of what it is Then follow the addition rules above EXAMPLES: (8) - (12) = (8) + (-12) = - and (-8) - (12) = (-8) + (-12) = -20 and (-8) - (-12) = (-8) + (12) = Notice the sign of the number in front of the subtraction sign never changes ••• Multiply or divide, then follow these rules to determine the sign of the answer • If the numbers have the same signs the answer is POSITIVE • If the numbers have different signs the answer is NEGATIVE • It makes no difference which number is larger when you are trying to determine the sign of the answer EXAMPLES: (-2)(-5) = 10 and (-7)(3) = -21 and (-2)(9) = -18 • NATURAL or Counting NUMBERS: {I, 2, 3, 4,5, , 11, 12, } • WHOLE NUMBERS: {O, 1,2,3, , 10, 11, 12, 13, } • INTEGERS: { , -4, -3 , -2, -1 , 0,1 , 2, 3,4, } • RATIONAL NUMBERS: {p/q I p and q are integers, q O}; 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: {xl 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 I x is the coordinate of a point on a number line}; the union of the set of Rational numbers with the set ofIrrational numbers equals the set of Real Numbers • IMAGINARY NUMBERS: {ai I a is a real number and i is the number whose square is -I }; i2 = -1; the sets of Real numbers and Imaginary numbers have no elements in common and are therefore disjoint sets • COMPLEX NUMBERS: {a + bi I a and b are real numbers and i is the number whose square is -I}; the set of Real numbers and the set of Imaginary numbers are both subsets of the set of Complex numbers EXAMPLES: + 7i and - 2i are complex numbers • TOTAL or SUM is the answer to an addition problem The numbers which are added are called addends EXAMPLE: In + = 14 , the and are addends and the 14 is the total or sum • DIFFERENCE is the answer to a subtraction problem The number that is subtracted is called the subtrahend The number from which the subtrahend is subtracted is called the minuend EXAMPLE: In 25 - = 17 , the 25 is the minuend, the is the subtrahend, and the 17 is the difference • PRODUCT is the answer to a multiplication problem The numbers that are multiplied are each called a factor EXAMPLE: In 15 x = 90, the 15 and the are factors and the 90 is the product • QUOTIENT is the answer to a division problem The number which is 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 ;- = , which may also be written as 5)43" or 4515, the 45 is the dividend, the is the divisor and the is the quotient • An EXPONENT indicates the number of times the base is multiplied by itself; that is, used as a factor EXAMPLE: In 53 the is the base and the is the exponent or power and 53 = (5)(5)(5) = 125, notice that the base, 5, was multiplied by itself times • PRIME NUMBERS are natural numbers greater than I that have exactly two factors, itself and one EXAMPLES: is prime because the only two natural numbers that multiply to equal are and 1; 13 is prime because the only two natural numbers that multiply to equal 13 are 13 and I • 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; is composite because 1, 3, and all multiply in some combination to equal • 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 because divides evenly into 12, into 30, and into 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, and is 12 because although 2, 3, and 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,the 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 the indicated division of the fraction EXAMPLE: In 3/4 , the is the numerator and also the dividend in the indicated division • Rule: Always divide by a whole number • If the divisor is a whole number simply divide and bring the decimal point up into the quotient (answer) 04 EXAMPLE: 4).16 • If the divisor is a decimal number, move the decimal point behind the last digit and move the decimal point in the dividend the same number of places Divide and bring the decimal point up into the 70 quotient (answer) EXAMPLE: 05) 3.501 - ; ' '" • This process works because both the divisor and the dividend are actually multiplied by a power often, that is 10, 100, 1000, or 10000 to move the decimal point 12 x lQQ = :l2J2 = 70 EXAMPLE: 05 x 100 DECIMAL NUMBERS I • 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 often EXAMPLE: In 324, the means 300 because it is times I ()2 (102 = 100) The means 20 because it is times 10 (10 I = 10), and the means times one because it is times 10° (10° = I) There is an invisible decimal point to the right of the In 5.82, the means times one because it is times 10° (10° = I), the means times one tenth because it is times 10- ( 10- = = 1110), and the means times one hundredth because it is times 10-2 (10- = I = III 00) PLACE VALUE 10° Ones or Units 10 Tens - 10' Hundreds · 103 Thousands -, I COMPLEX NUMBERS Real Numbe:cs Rational Integers Whole [SJ" ~ ; 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