NUMBER SYSTEMS The natural numbers are the numbers we count with: 1, 3, 4, 5, 6, , 27, 28, The whole numbers are the numbers we count with and zero: 0, 1, 2, 3, 4, 5, 6, The integers are the numbers we count with, their negatives and zero: , -3, -2, -1, 0, I, 2, 3, -The positive integers are the natural numbers -The negotfve integers are the "minus" natural numbers: -1, -2 -3, -4, The rcmonal numbers are all numbers expressible as ~ fractions The fractions may be proper (less than one; Ex: k) or improper (more than one; Ex: ft) Rational numbers can be positive (Ex: 5.125 = rational: Ex: = y ¥-) or negative (Ex: - ~) All integers are The real numbers can be represented as points on the number line All rational numbers aTe real, but the real number line has many points that are "between" rational numbers and aTe called ilTotional Ex: 0, To, v'3 - Real numbers , -! 1.25 etc -1, -2, -3, 9, 0.12112111211112 Irrationals The imaginary numbers are square roots of negative numbers They don't appear on the real number line and are written in terms of i = FI Ex: J=49 is imaginary and equal to iV49 or 7i v'5~ The complex numbers are all possible sums of real and imaginary numbers; they are written as a + bi where a and bare real and i = FI is imaginary All reals are complex (with b = 0) and all imaginary numbers are complex (with a = 0) The Fundamental Theorem of Algebra says that every Integers ~~ - - Whole Dwnbers Natural numbers )'t2+3 etc Venn Diagram of Number Systems polynomial of degree n has exactly n complex roots (counting multiple roots) I SETS A set Is ony collecllon-finite or infinite :: (; SttS c 84 Sto c( ~ -a is a real number a+ (-a) =(-a)+a=O Also, -(-a) = a a + b is a real number is a real number a-l=l-a=a is the multiplkotive identity If a i- 0, ~ is a real number + c) = a b + a c + c) a = b· a + c· a Inequality « and» Property Equality (=) Reflexive a=a Symmetric If a = b, then b = a Ifa~bandb=c, Transitive INEQUALITY SYMBOLS Meaning Other properties: Suppose a, b, and c are real numbers then a = c t = a a b is a real number 11 a < band b < c, then a < c Example < less than < and < 56 > greater than > and 56 > i not equal to i and -1 i ~ less than or equal to ~ and ~ 2: greater than or equal to 2: and 2: - 29 The shorp end aiways points toward the smoller number; the open end toword the larger Addition and subtraction If a ~ b, then a+c=b+cand a-c=b-c Multiplicotion and division b, then ac = be and ~ = ~ (if c i 0) If a ~ 11 a < b, then a+ c < b+ c and a-c< b-c If a < band c > 0, then ac < be and %< ~ If a < band c < 0, then switch the PROPERTIES OF EQUALITY AND INEQUALITY u·~=~·a=l Also, (b There are also two (derivative) properties having to with zero Mullipllcotion by zero: a = a = O Zero product property: If ab = then a = or b = (or both) Sign + b) + c = a + (b + c) a· (b· c) = (a· b) c Assoclotive a (b inequality: ac > be and ~ > ~ Trichotomy: For any two rea) numbers a and b, exactly one of the following is true: a < b a = b, or a > b LINEAR EQUATIONS IN ONE VARIABLE A linear equation in one variable is an equation that, after simplifying and collecting like terms on each side will look iike a7 + b = c or like IU" b ("J" + d Each Side can Involve xs added to real numbers and muihplied by real numbers but not multiplied by other rS Ex: 1(-~-3)+.c=9-(7-~) is a Iineor equation j" = and 7(X + 4) = and Vi = ore not linear But in one vW'iable will always have (a) exactly one real number solution, (b) rlO solutions, or (c) all real numbers as solutions FINDING A UNIQUE SOLUTION J(-t -3) TT=9-(x- real numbers are solutions Add 8x to both sides to get 5x + 8x - 18 = 77 or 13x-18=77 Add 18 to both sides to get 13x = 77 + 18 or 13x = 95 Divide both sides by the variable's coefficient Stop if a = U Does ~ (2.~~ Get rid of fractions outside parentheses - 3) + ~ = - (~- ~)? Yes! Hooray II~\I :C.ll,., II' j Ifill~I.] ~\ 14f'.'il'.' :JII Use the same procedure as for equalities, except flip the inequality when mliltiplying or dividing by a negative number Ex: -x > is equivalent to x < -5 -The inequality may have no solution if it reduces to an impossible statement Ex: :£ + > x + reduces to > -The inequality may have all real numbers as solutions if it reduces to a statement that is always true Ex: - x 2: - x reduces to ~ and has infintely many solutions Solutions given the reduced Inequality and the condition: Multiply through by the LCM of the denominators Ex: Multiplyby4 toget3(-t -3) +4x=36-4(x- ~) Simplify using order of operations (PEMDAS) DETERMINING IF A UNIQUE SOLUTION EXISTS Use the distributive property and combine like terms on each side Remember to distribute minus signs - Multiply by to get rid of fractions: 5x - 18 = 77 - 8x Move variable terms and constant terms to different sides Usually, move variables to the side that had the larger variable Divide by Ij to get x = ~ Check the solution by plugging into the original equation ~) Ex: Distribute the left-side parentheses: -~x - + 4x = 36 - (x - ~) Combine like terms on the left side: ~x Any linear equation can be simplified into the form ax = b for some a and b If a i O then x = ~ (exactly one solution) If a = but b i 0, then there is no solution If a = b = 0, then all coefficient to begin with Equation should look like a::1: = b Linear eCJlUlh'ons Ex: Distribute the right-side parentheses; ~x - = 36 - 4x + ~ Combine like terms on the right side: ~x - = ¥ - 4x Repeat as necessary to get the form ax + b = ex + d = 36 - (x - ~) a>O -The original equation has no solution if, after legal transformations, the new equation is false Ex: = or 3x - = + 3x -All real numbers are solutions to the original equation if, after legal transformations, the new equation is an identity Ex: 2x = 3x -.r or = a b} OR {b ~ and a < -b} Ex: IIOI + II < 7x + is eqUivalent to 7.e + ~ and -7x - < lOx + < 7x + Thus the equations 7x + ~ 0, -7x - < lOx + I and IOI + I < 7x + must all hold SolVing the equations we see that interval and every boundary point The point x = is often good to test Ifs simplest to keep track of your information by graphing everything on the real number line Ex: IlOx + 11 < 7x + Solve the three equntions lOx + I = 7I + -lOx-l=7x+3 and 7x+3=0 to find potentinl boundary points The three points are not surprisingly ~ and - ~ Testing the three boundary pOints and a point from each of the four intervals gives the solution - ~ $ x < ~ At, GRAPHING ON THE REAL.: NUMBER LINE The real number line is a pictorial representation of the real numbers: every number corresponds to a point Solutions to one· vorioble equations and (especiallyi inequalities may be graphed on the real number line The idea is to shade in those ports of the line that represent solutions Origin: A special point representing O By x ::; a: Shaded closed ray: everything to the left of and including a o Open (ray or interval): Endpoints not oa I Closed (ray or interval): Endpoints included GRAPHING SIMPLE STATEMENTS x = a: Filled-in dot at a o I a • ; o I a • I a+b both inequalities Shade the portions that I would be shaded by both if graphed independently I b Ix - al < b; I.e - "I ::; b: The distance from a to x is less than (no more than) b; or x is closer than b to a Plot the interval (open or closed) a - b < x < ,,+ b (or" - b::; x ::; a + b.) '# a: Everything is shaded except for a around which there is an open circle o '­ : Filled-in circle if the endpoint is included, a I: I a+b : I b • I a is really x < -3 OR x > Equivalently, it is{x, x < -3} U {x : x > 5} The graph is the union of the graphs of the individual inequalities Shade the portions that would be shaded by either one (or both) if graphed independently -Endpoints may disappear Ex: x > OR just means that x > The point is no longer an endpoint x ~ b b a ~ "0 V> '" N ~.!!l ~ c :E ~~.c~! S j; ~ ~ 8- j!! ~ ~~ ~ul o::j ~~~~~ 'c"5~ ~.;:: is Graph both equations on the same Cartesian plane The intersection of the graph gives the simultaneous solutions (Since points on each graph correspond to solutions to the appropriate equation, points on both graphs are solutions to both equations.) -Sometimes, the exact solution can be determined from the graph; other times the graph gives an estimate only Plug in and check -If the lines intersect in exactly one point (most cases), the intersection is the unique solution to the system 1x + , u sri ~ ~ 5ro ~ & x exoclly one solution -4 - If the lines are parallel, they not intersect; the system has no solutions Parallel lines have the same slope; if the slope is not the same, the lines will intersect Ex:{ y x -4 no solutions x ­ 4y = 2x-11=2y Using the first equation to solve for y in lerms of x gives y = t(x -1) Plugging in 10 the second equation gives 2.T -11 = O(x ­ 1») SolVing for x gives x = Plugging in for y gives y = HI - 1) = ~ Check that (7,~) works -4 Express both equations in the same form ax + by = c works well -If the lines coincide, there are infinitely Look for ways to add or subtract the many solutions Effectively, the two equations to eliminate one of the variables equations convey the same information ' -If the coefficients on a variable in the two y equations are the same, subtract the '2 y equations -If the coefficients on a variable in the two equations differ by a sign, add the v = equations -4 -If one of the coefficients on one of the variables (say, x) in one ofthe equations is x 1, multiply that whole equation by the x­ infinitely many coefficient in the other equation; subtract solutions the two equations -If no simple combination is obvious, simply pick a variable (say, x) Multiply the first equation by the x-coefficient of the second equation, multiply the second equation by the x-coefficient of the first -Use one equation to solve for one variable equation, and subtract the equations (say, y) in terms of the other (x): isolate y If all went well, the sum or difference equation on one side of the equation is in one variable (and easy to solve if the -Plug the expression for y into the other original equations had been in ax + by = c equation fonn) Solve it -Solve the resulting one-variable linear -If by eliminating one variable, the other is equation for x eliminated too, then there is no unique -If there is no solution to this new equation, solution to the system If there are no there are no solutions to the system solutions to the sum (or difference) -If all real numbers are solutions to the equation, there is no solution to the new equation, there are infinitely many system If all real numbers are solutions to solutions; the two equations are the sum (or difference) equation, then the dependent two original equations are dependent and -Solve for y by plugging the x-value into the express the same relationship between the expression for y in terms of x variables; there are infinitely many -Check that the solution works by plugging it solutions to the system into the original equations -Plug the solved-for variable into one of the original equations to solve for the other variable Ex.{ x -4y = 2x-ll=2y Rewrite to get { X - 4y = 2x _ 2y = 11 The x-coefficient in the first equallon is I, so we multiply the first equation by 210 get 2x - 8y = 2, and subtract this equation from Ihe ariginal second equation to get: (2 - 2)x + (-2 - (-8))y = 11 - or 6y = 9, which gives y = ~, as before CRAMER'S RULE The solution to the simultaneous equations aX+by=e { ex + dy = f if ad - be , l~i[.I;I:a is given by o j: rl1~' I X=~ ad-be Y !l.=.E! (ul-be i'~I.l!ll';j r1:J Ii l There is a decent chance thol a system of Iineor equalions has a unique solution only if there are as mony equations as variables -If there are too many equations, then the conditions are likely to be too restrictive, resulting in no solutions (This is ollly actually true if the equations are "independent"-each new equation provides new information about the relationship of the variables.) -If there are too few equations, then there will be too few restrictions; if the equations are not contradictory, there will be infinitely many solutions -All of the above methods can, in theory, he used to solve systems of more than two linear equations In practice, graphing only works in two dimensions It's too hard to visualize planes in space -Substitution works fine for three variables; it becomes cumbersome with more variables -Adding or subtracting equations (or rather, arrays of coefficients called matrices) is the method that is used for large systems EXPONENTS AND POWERS l;jll!i-j']lja:X,,~14~'if Exponential notation is shorthand for repeated multiplication: 3·3= 3' and (-2y)· (-2y)· (-2y) = (-2y)' In the notation a'l, a is the base, and 11 is the exponent The whole expression is "a to the nth power," or the "nth power of a, or, simply, "a to then." Quotient of powers: am ~ = a m - n (-a)" is not necessarily the same as -(a") Exponentiation powers: (am)" = a~" To raise a power to a power, multiply e.xponents Ex: (-4)' = 16, whereas -(4') = -16 Following the order of operation rules, Quotierrt of a product: Negative powers: a- a" (Iia)" = t;; Exponentiation distributes over multiplication and division, but not over addition or subtraction Ex: (2xy)' = 4x'y', but (2+x +y)' '" + x' +y' If the bases of two powers are the same, then to divide, subtract their exponents a is "u squared;" a3 is un cubed." -a" Power of a product: (ab)" = a"b" Product of powers: oman = am + n If the bases are the same, then to multiply, simply add their exponents Ex: 23 28 = 11 Zeroth power: aO = To be consistent with all the other exponent rules, we set aO = unless a = O The expression 00 is undefined n = ! ­ a" We define negative powers as reciprocals of positive powers This works well with all other rules Ex: 23 2- = ~ = Also, 23 2-' = 23 +( -3) = 2° = Fractional powers: a ~ = if