PREALGEBRA REVIEW DEFINITIONS

Transcription

1 1. Subtractio is the iverse of additio: A. If a - b c, the c + b a B. So if 10-7, the Divisio is the iverse of multiplicatio: A. If a b c, the c b a B. So if , the PREALGEBRA REVIEW DEFINITIONS. Additio ad multiplicatio follow the commutative properties: A. a + b b + a, so B. a b b a, so Affiig zeroes to the left of a umber or to the right of a decimal poit will ot chage the umber: A. So , ad so o. B. Also, , ad so o. C. However, ote: 84 does ot equal 804, 4.7 does ot equal 4.07, ad 5. does ot equal To multiply a umber by 10, 100, ad so forth, simply attach the appropriate umber of zeroes to the right of the umber, or simply move the decimal poit the appropriate umber of places to the right. A. So , ad , 500 B. Also, , ad To divide a umber by 10, 100, ad so forth, simply move the decimal poit of the umber the appropriate umber of decimal places to the left: A. So , ad Whe dividig, you may cacel a equal umber of right-had zeros: A. So , ad 8, , A PERCENT (%) meas parts of % meas 6 parts of 100, or 6/100. To covert to a percet, just multiply by 100. If the umber is a decimal, use the rule from #6 to multiply by 100 ad attach the % sig X 100 2% To covert from a percet to a decimal, just drop the % sig ad divide by % 2/ ORDER OF OPERATIONS (PEMDAS) Please: do all operatios withi paretheses ad other groupig symbols (such as [ ], or operatios i umerators ad deomiators of fractios) from iermost outward. Ecuse: calculate epoets My Dear: do all multiplicatios ad divisios as they occur from left to right Aut Sally: do all additios ad subtractios as they occur from left to right. Eample: (8-6) 2 Epressio i paretheses gets calculated first (2) 2 Net comes all items with epoets (4) Net i order comes multiplicatio. Multiplicatio ad Divisio always come before additio or divisio, eve if to the right Now whe choosig betwee whe to do additio ad whe to do subtractio, always go from left to right, so do 20-2 first, because the subtractio is to the left of the additio Now fially we ca do the additio The Test of Reasoableess I geeral applyig a test of reasoableess to a aswer meas lookig at it i relatio to the umbers operated upo to determie if it s i the ballpark. Put i simple terms, you look at the aswer to see if it makes sese. For eample, if you determie 10% of $75 to $750, you should immediately otice somethig is very wrog, because 10% of somethig is much smaller tha the origial amout. The test of reasoableess comes very much ito play i word problems. For eample, questios askig for legth, dollars, or time should ever give egative aswers because those thigs would ot make sese if they were egative (what is - feet?). Also, thik about what the aswer should be. If I ivest $400 dollars i a bak at 8% iterest for years, I would epect the balace to be larger tha the origial amout that I ivested (that s why we put moey i the bak!), so a aswer that is smaller tha $400 is obviously wrog.

2 Factor - Oe of two or more quatities that divides a give quatity without a remaider. For eample, 2 ad are factors of 6; a ad b are factors of ab. Factor X Factor Product Product - The umber or quatity obtaied by multiplyig two or more umbers together. Factor X Factor Product Divided - A quatity to be divided. Divisor - The quatity by which aother quatity, the divided, is to be divided. Quotiet - The umber obtaied by dividig oe quatity by aother. I 45 15, 15 is the quotiet. Divided/ Divisor Quotiet or Divided Divisor Quotiet divisor quotiet divided Remaider - The umber left over whe oe iteger is divided by aother: The remaider plus the product of the quotiet times the divisor equals the divided. If there is a remaider, the Divided Quotiet X Divisor + Remaider Least Commo Multiple (LCD) - The smallest quatity that is divisible by two or more give quatities without a remaider: 12 is the least commo multiple of 2,, 4, ad 6. Also called lowest commo multiple. Greatest Commo Factor (GCF) - The largest umber that divides evely ito each of a give set of umbers. The greatest commo divisor is useful for reducig a fractio ito lowest terms. Cosider for istace where we cacelled 14, the greatest commo factor of 42 ad 56. Prime umber is a whole umber that has eactly two factors: itself ad 1. Composite umber is a whole umber that has more tha two factors.

3 A set is a collectio of objects. The objects i the set are called the elemets of the set. The roster method of writig sets ecloses a list of the elemets i braces. Eample: The set of eve atural umbers less tha 10 ca be writte like this: {2,4,6,8}. The set of atural umbers is {1,2,,4,5,6,7,.}. These are basically the coutig umbers. The set of whole umbers is the set of atural umbers ad the umber, 0. {0, 1, 2,, 4, 5,.} The set of itegers is the set of whole umbers ad their opposites. {. -4,-,-2,-1,0,1,2,,4,.} The umber 0 is a iteger, but it is either egative or positive. For ay two differet places o the umber lie, the iteger o the right is greater (>) tha the iteger o the left. The absolute value (usig the symbol) of a umber is its distace from zero o the umber lie. The absolute value of a umber is ALWAYS POSITIVE (or 0). 5 5, -5 5, , is 4 uits away from is 4uits away from 0

4 Addig ad Subtractig Itegers Whe addig two itegers with the same sig, just igore the sigs, the attach them o the aswer ( + 5) -8 Whe addig two itegers with differet sigs, take the absolute values (make both umbers positive), ad subtract the smaller oe from the larger oe. The sig of the iteger with the larger absolute value will be the sig of your aswer. 7 + (-8) The absolute value of 7 is 7 7 The absolute value of -8 is -8 8 larger Remember i the origial problem, the iteger whose absolute value was 8 was -8, so our aswer is egative. 7 + (-8) -1 Subtractig a egative iteger is the same as a iteger. (-5) (-) Subtractig a positive iteger is the same as a iteger (-) (-) 1 Multiplyig ad Dividig Itegers (positive iteger) X (positive iteger) ( iteger) (positive iteger) (positive iteger) ( iteger) (egative iteger) X (egative iteger ) ( iteger) (egative iteger) (egative iteger ) ( iteger) (positive iteger) X (egative iteger) ( iteger) (positive iteger) (egative iteger) ( iteger) (egative iteger) X (positive iteger ) (egative iteger) (egative iteger) (positive iteger ) (egative iteger)

5 Memory Tip: Whe somethig bad happes to a good perso, that s bad. Whe somethig good happes to a bad perso, that s bad. Whe somethig good happes to a good perso, that s good. Whe somethig bad happes to a bad perso, that s good. Whe a egative iteger is raised to a eve power, the result is. (-2) 4 (-2)(-2)(-2)(-2) 16 Whe a egative iteger is raised to a eve power, the result is. (-2) 5 (-2)(-2)(-2)(-2)(-2) -2 Epoets are oly applied to the umber directly diagoally left of it. If a egative iteger is raised to a power, it must be i paretheses () i order for the epoet to apply to the egative umber. (-2) 2? (-2) 2 (-2)(-2) (2)(2) -4

6 Epoet - A umber or symbol, as i ( + y), placed to the right of ad above aother umber, variable, or epressio (called the base), deotig the power to which the base is to be raised. Also called power. The epoet (or power) tells how may times the base is to be multiplied by itself. Eample 1: ( + y) ( + y)(+y)(+y) Eample 2: (-) 4 (-)(-)(-)(-) 81 Properties of Epoets If m ad are itegers, the m m+ If m ad are itegers, the ( ) m m If is a real umber ad 0, the 0 1 If m,, ad p are itegers, the ( y) y, ad ( y ) y m p m p p If m ad are itegers ad 0, the m m If is a positive iteger (- is egative), ad 0, the 1 ad 1 If is a iteger ad b 0, the a a b b If is a positive iteger (- is egative), ad b 0, the a b b a

7 FRACTIONS, DECIMALS, ad PERCENTS Numerator A fractio is just a divisio problem. Deomiato r PREALGEBRA REVIEW DEFINITIONS Numerator Deomiator A fractio is i LOWEST TERMS whe the umerator ad deomiator have o commo factors is ot i lowest ter ms, Mied Number - A umber, such as umber ad a fractio. is i lowest ter ms 5 6, cosistig of a iteger ad a fractio. A mied umber is just the sum of a whole 5 Improper Fractio - A fractio i which the umerator is larger tha the deomiator. Covertig from mied umber to improper fractio: Mied umbers must be coverted to improper fractios or decimals before doig ANY MULTIPLICATION OR DIVISION OPERATIONS o them. 5 A fractio ca be coverted ito a decimal by dividig: Numerator Deomiator deomiato r umerator Whe ADDING or SUBTRACTING FRACTIONS, they must have the same deomiator, the you just add the umerators ad leave the deomiator the same The deomiators, 6 ad 8, are ot the same, so we must fid the LEAST COMMON DENOMINATOR to covert these fractios ito equivalet oes with the same deomiator. The LCM of 6 ad 8 is the SMALLEST NUMBER THAT BOTH 6 ad 8 ca go ito. Choose the larger deomiator (which is 8 i this case) ad start takig multiples util 6 ca go ito it. Does 6 go ito 8? NO Does 6 go ito 82? NO Does 6 go ito 8? Yes, 6 goes ito 24. Therefore 24 is the LCM. Multiply the umerator ad deomiator of each fractio by whatever it takes to get the LCM as the ew deomiator. The oce you have the same deomiators i each fractio, just add the umerators ad leave the deomiator the same MULTIPLYING FRACTIONS Multiplyig Fractios uses a differet rule. Whe multiplyig fractios, you multiplyig the umerators AND the deomiators This fractio ca simplified by cacelig out commo factors i the umerators ad deomiators (tops ad bottoms)before multiplyig across. 6 ca be rewritte as 2, ad the s ca be cacelled out DIVIDING FRACTIONS The rule for dividig fractios is simple. Just take the RECIPROCAL(flip the top ad bottom) of the DIVISOR (the fractio to the right of the symbol) multiply

8 PERCENT part of a hudred %. 05 Whe covertig from Percet to a Decimal, drop the % sig ad move the decimal place to the LEFT TWO PLACES. 65% 6.5 Whe covertig from a Decimal to a Percet, move the decimal place to the RIGHT TWO PLACES ad the attach a % sig to the ed % Whe covertig a fractio to a percet, first covert the fractio to a decimal (see previous page). Percet Problems: Amout Percet * Base 4 is 40% of 10 Eample 1: What is 0% of 60? Amt Percet X Base Amt 0% X 60 Amt Covert 0% to decimal. 0%.0 Amt.0 X Eample 2: 200% of what is 400? 200% is the Percet Covertig Percet to a Decimal gives 200% is the Amout What is the Base. Base Amt / Percet 400 / Percet Base Eample : is what percet of 45? Amt Base 45 Percet Amt/ Base Percet / Covertig to Percet usig % symbol gives % which rouds to 6.67%

9 GEOMETRY - itercept - poit where graph crosses -ais. y 0 at this poit. y-itercept poit where graph crosses y-ais. 0 at this poit. slope rise/ru of a lie. If liear equatio is i the form y m + b, m is the slope, ad b is the y-itercept. y - ½ +1 straight agle 180 degrees. Adjacet agles that form a straight agle add up to 180 degrees. A B A + B 180 right agle 90 degrees. Complemetary agles area adjacet agles that form a right agle ad add up to 90 degrees. acute agle less tha 90 degrees obtuse agle greater tha 90 degees. Vertical agles (opposite agles formed from itersectig lies) are cogruet (the same). a b c a 2 +b 2 c 2 Pythagorea Theorem: The sum of the squares of the legs of a right triagle is equal to the square of the hypoteuse. QUADRILATERALS (4-sided polygos): POLYGONS Parallelogram Rhombus Rectagle Square Trapezoid (oly 2 sides are parallel) (opposite sides parallel) (all sides same size) (each agle 90 ) (each agle 90 ad all sides same size) Triagles Scalee Triagle Isosceles Triagle Equilateral Triagle Right Triagle (o sides the same) (2 sides the same) ( sides the same) (oe agle is 90 ) Agles of a triagle add up to 180 degrees. Perimeter distace aroud the edges of a object Perimeter of a square 4s, where s legth of oe side Perimeter of a rectagle 2W + 2L, where W width ad L legth Perimeter of a triagle side 1 + side 2 + side Area amout of surface covered by a object. Area of a square s 2 Area of a rectagle WL Area of a triagle ½ bh, where b legth of base ad h height Area of a parallogram bh b legth of base ad h height

10 CIRCLES: A circle is the set of all poits that are a fied distace from the ceter. The fied distace is called the radius, ofte abbreviated by the variable, r. The diameter starts at oe side of the circle, goes through the ceter ad eds o the other side. So the Diameter is twice the Radius: Diameter 2 Radius 2r Circle Sector ad Segmet Slices There are two mai "slices" of a circle. Ceter The "pizza" slice is called a Sector. Give this property, you ca make a formula for the circumferece by multiplyig both sides of this equatio by the diameter. Circumferece (Diameter) π ( Diameter) Diameter Circumferece π (Diameter ) Circumferece π (2r) 2πr Area of a Circle: Ad the slice made by a chord is called a Segmet. Commo Sectors The Quadrat ad Semicircle are two special types of Sector: Quarter of a circle is called a Quadrat. Half a circle is called a Semicircle. If you make a parallelogram out of a circle by piecig together the slices of pie, you will use half the circumferece for oe base ad half the circumferece for the other base (so base ½(2pr)pr). The height of the parallelogram will be the radius, r. Now we ow the area of a parallelogram is base X height. So the area of a circle is (pr)(r). Area of a Circle pr 2