Best Free CLEP College Math Study Guide

Pythagorean theorem the Pythagorean theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse of a right triangle the legs of a right triangle are the two sides that form the right angle the hypotenuse is the side that's across from the right angle let's look at two examples find the missing sides of the right triangles on this first example we're given the lengths of the two legs of the right triangle and what we need to find is our hypotenuse the hypotenuse again is the C in the Pythagorean theorem so what we're trying to find is C and we'll start by writing the Pythagorean theorem a squared plus C squared is equal to C squared a can be either one of the legs and B can also be either one of the legs so four squared plus three squared is equal to and the hypotenuse is what we don't know so we'll leave it as C our variable that we're trying to find then we simplify four squared is sixteen plus three squared is nine is equal to C squared then we need to combine like terms 16 plus 9 is 25 25 is equal to C squared we're almost there but we haven't found C yet right now we know what C squared equals and that's 25 but to find C we need to do the opposite of squaring which is square rooting so we square root both of our sides the square root of 25 is 5 and the square root of C squared is C which means that our hypotenuse is 5 on this next triangle we know the hypotenuse this time across from the right angle is our hypotenuse and that's given to us as 17 so what we're trying to find now is one of our legs and this leg can either be your a or your B so I'm going to call it my B again we'll start with Pythagorean theorem a squared plus B squared is equal to C squared a and B are your legs and C is your hypotenuse so we know one of our legs 15 so I'll substitute that for a squared plus we're trying to find our other leg B squared we don't know it equals and then our hypotenuse which is our C is 17 of 17 squared now we need to simplify 15 squared is 225 plus B squared is equal to 17 squared is 289 since we're trying to solve for B that means we need to get be alone and to do that we're going to have to subtract 225 from both sides 225 minus 225 is 0 so on the left side we're left with B squared is equal to and then 289 minus 225 9 minus 5 is 4 8 minus 2 is 6 and 2 minus 2 is 0 so again we found B squared but we haven't found B yet so to solve for B we need to take the square root of both sides the square root of B squared is B and the square root of 64 is 8 so that means that our other leg on our right triangle is 8 flexion a reflection is like a mirror image in a reflection all the points in the figure are reflected across the same line this line may or may not pass through the figure itself let's look at an example of a reflection we're going to reflect triangle ABC about the y-axis so we're going to draw a mirror image of this triangle on the other side of the y-axis let's start with point a point a is at negative 2/5 so when we reflect it about the y-axis it'll be in the same place on the opposite side of the y-axis so on the other side it will be at positive 2/5 this is our new point a B is at negative 5 1 so when we reflect it on to the other side it'll be in the same place on the opposite side of the y-axis so instead of it negative 5 1 it'll be at positive 5 1 and then finally C is that negative 1 1 so when we reflect it it's at positive 1 1 so you may have noticed that the Y values of our numbers stayed the same while our X values change to just the opposite we can connect our three points together and there we have a reflection of triangle ABC about the y-axis right triangle word problem a ramp with a 30 degree angle from the ground is to be built up to a two foot high platform what will be the length of the ramp how far from the platform will it extend to the nearest inch with problems like these it's very helpful to draw a picture so we have a ramp that makes a 30-degree angle with the ground so let's start with that here's the ground here's our 30-degree angle and it's to be built up to a two foot high platform so then here we have a platform and it's two feet high above the ground there's a right angle there perpendicular to the ground the first question is what will be the length of the ramp so this is what we're trying to find well before I start to find X I first want to fill in what this angle is you might already know that since we have a 30-degree angle and a 90-degree angle this angle has to be 60 degrees but if you didn't know that you could find it because there are a hundred eighty degrees in a triangle so if you have thirty to ninety you get 120 180 minus 120 leaves you with sixty degrees for that third angle now that we know it's a 30-60-90 triangle we can apply our 30-60-90 rules to finding the length of our ramp our ramp is across from the 90 degree angle therefore that's our hypotenuse the side we know the likely nose across from the 30-degree angle so it's our shorter leg and the hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg our hypotenuse again we're using x four equals two times the shorter leg which is that two foot side so two times two feet which means X is four feet therefore the length of the ramp is four feet so we've answered the first question then they asked how far from the platform will it extend how far from the platform will the ramp extend so now they're asking us to find that length and Y is across from our 60-degree angle so that's our longer leg and the longer leg of a 30-60-90 triangle is the square root of three times the shorter leg our longer leg again is y equals the square root of three times and our shorter leg is still that two foot length so Y is two square roots of three feet they didn't ask us for the answer and feet they asked to the nearest inch so the first thing I'm going to do is convert two square roots of three feet into inches by multiplying by 12 so Y is two square roots of three times twelve since there are 12 inches in one foot which is 24 square roots of three inches they didn't ask for it in simplest radical form so then we'd actually want to calculate what 24 square roots of three inches is and it's about 42 inches therefore how far from the platform will the ramp extend the ramp will extend about 42 inches from the platform there we have it rotation a rotation is when you do a transformation where you turn a figure about a given point a certain amount of degrees now if you rotate something zero degrees then you haven't moved it at all but otherwise the whole shape is going to move about that point that you rotate it and you can rotate it as about a point on your shape in your shape or even outside of your shape so here we're going to rotate figure ABCDE 90 degrees about point D so since we're rotating it about point D that means that D will remain fixed it's still going to be right there after we rotate it that's still where our point D will be but the rest of our shape is going to turn 90 degrees and it's you can think of this kind of as like the pivot point so the rest the whole rest of our shape is going to turn 90 degrees so it's going to come down here now and this will be where your point B is now about right there and that'll come down here to your point a now and we'll come across here to your point E and I drew that a little too far over it should be right here about two point D and then this goes up and in and there's your point C so there it is rotated 90 degrees you translation a translation is just a slide so it's where you take a shape or a figure and you slide it and so generally all of your points are going to move unless you're not moving it at all so here we're going to translate trapezoid ABCD one unit right and two units down so that means we're going to slide this trapezoid and thus each one of its points one unit to the right and two units down let's start with point a point a is that the ordered pair negative three zero and we need to move it one unit to the right so that would be at negative two and then two units down so we would now be at negative 2 negative 2 for our slides so negative 2 negative 2 that's where our new point a is after we've moved it one unit to the right and 2 units down we're at negative 2 negative 2 and next we'll move B you can move any of the points I'm just going to go in alphabetical order so next I'm going to take point B and B is at negative 1/3 and I'm going to slip slide it or translate it one unit to the right so is it that negative one now I'm going to move it one unit to the right so it'd be on 0 and then two units down we'd be at one so over 0 up one is our new point for B now for C C is at 1/3 and again I'm going to move it one unit to the right so I'm moving it from one over 1/2 to one unit to the right and then two units down so from here it's moving to down so it would be at 2 1 / 2 up 1 that's our new position of C and then finally we have D and D is at 3 0 so again I'm going to move it one unit to the right so from 3 it'll move over to 4 and then I need to move it 2 units down so it's at 4 and I'm going to move it down to negative 2 so over 4 and down to negative 2 there's our point D and we have our new trapezoid that has been translated one unit to the right and two units down so you can see it's still the same shape it's just been moved in our coordinate plane laws of exponents we're going to look at seven different laws the first is that any number raised to the first power is just itself if you'll remember what an exponent is it tells you how many times to multiply the base number times itself so if you're multiplying the base number times itself well just one time then that's just that number like 2 to the first power would just be 2 or negative 3 to the first power would just be negative 3 the second rule says that 1 raised to any power is 1 because all you're doing is multiplying it 1 times itself and no matter how many times you multiply 1 times itself it'll always be 1 so for instance like 1 to the third power that means 1 times itself 3 times 1 times 1 is 1 times 1 is 1 so it doesn't matter how many times you multiply 1 times itself you'll always get 1 the third law says that any number raised to the 0 power is 1 this is one of my personal favorites I think it's pretty cool like 8 raised to the 0 power it's just 1 or negative 10 raised to the 0 power again just one or even say like 1/2 raised to the 0 power still 1 any number raised to the 0 power is always 1 the fourth law is when you're multiplying with the same bases so notice a and a our bases are the same but we have different exponent when you multiply numbers with the same bases you add the exponents let's look at why so say we were doing 2 cubed times 2 to the 4th well what that really means is 2 cubed is 2 times itself 3 times so 1 2 3 times times 2 times itself 4 times so 1 2 3 4 times so we've multiplied 2 times itself a total of seven times so instead of going through all of this we can just use that rule that when we multiply numbers with the same bases we simply add the exponents and then you could simplify from there the fifth rule is pretty similar to the fourth rule except we're dividing numbers at the same bases so instead of adding our exponents we subtract our exponents for example if we were doing 2 to the fifth divided by 2 cubed what we're really doing is 2 times 2 times 2 times 2 times 2 divided by 2 times 2 times 2 so that's 2 to the fifth 2 times itself 5 times divided by 2 times itself 3 times well we can simplify by canceling 2 divided by 2 is 1 2 divided by 2 is 1 and 2 divided by 2 is 1 so what we're left with in is simply 2 squared or 4 so instead of doing this we can use our rule that when we're dividing numbers with the same bases we simply subtract the exponents and that saves us a little bit of time 2 to the fifth divided by 2 cubed 5 minus 3 is so it's 2 squared which is 4 it also works if your denominator the exponent on your denominator is larger let's look at one of those so that'd be like 2 squared divided by 2 cubed so you still subtract your exponents so it's 2 squared minus 3 which is 2 to the negative 1 and then you would simplify that by bringing this to your denominator 1/2 the 6th rule says that when you raise a power to a power like a to the n raised to the M you multiply your exponents so for instance with 3 squared cubed what that really means is 3 squared times itself 3 times so that's 3 squared times 3 squared times 3 squared 3 squared times itself 3 times 3 squared times 3 squared times 3 squared is 3 to the sixth so all you have to do is multiply your exponents 2 times 3 is 6 3 to the sixth the seventh rule says that when you have operations inside parentheses that are raised to an exponent then each term inside those parentheses is raised to that exponent so for instance like 2 times 3 squared that is 2 squared times 3 squared and again you could simplify from there left off my in there that's important now on this next one same thing just with division so a divided by B that quantity raised to the N would be a to the end by be to the end so everything in the parentheses is raised to the N power so it'd be like doing 4/3 cubed so it'd be 4 to the 3rd divided by 3 to the 3rd and then again you could simplify and those are our laws of exponents absolute value absolute value is the distance of a number from zero and is always positive it's denoted by a pair of vertical lines surrounding the value such as the absolute value of three you can think of absolute value like you're being asked a question and the question is how far is three from zero three is one two three places from zero so the absolute value of 3 is 3 the absolute value of a negative number and it's positive counterpart are the same so the absolute value of negative 3 is also 3 because negative 3 is also 3 places from 0 1 2 3 places from 0 the absolute value of a difference when you're doing that the order doesn't matter let's look at the absolute value of a couple of differences the absolute value of 8 minus 3 and the absolute value of 3 minus 8 when you're doing the absolute value of a difference again the order doesn't matter the result will still be the same let's see how that happens start on the inside finding your difference 8 minus 3 is 5 so we have the absolute value of 5 and again absolute value is the distance from zero 5 is 5 places from 0 now we'll look at the difference switched around 3 minus 8 3 minus 8 you could do add the inverse of 3 plus an egg of eight three plus a negative 8 is negative five so we have the absolute value of negative five and like we talked about over here the absolute value of a number and it's a negative counterpart are the same so the absolute value of five and negative five is the same because again absolute value is the distance from zero and distance can never be negative so the absolute value of negative five is also five since negative five is five places from zero adding and subtracting integers when adding two positive integers together the rules are simple you add them and you'll always get a positive result three plus four is seven when adding two negative integers together you add them the same as you do when they're positive but then you change the sign to negative since they're both negative three plus four is seven since both of these numbers are negative our result is negative when you're adding two integers with different signs you're going to subtract you're going to subtract the number with the smaller absolute value so in this case five from the number with the larger absolute value in this case seven and then you'll keep the sign of the number with the larger absolute value let's try it seven minus five is two and since negative seven has a larger absolute value and it's negative the result is also negative when you're subtracting you can change a subtraction problem to an addition problem and then follow the same rules you do for addition to change this to an addition problem you're going to first change subtraction to addition and then we're going to reverse the sign on the second term so this two becomes a negative two notice that or note that minus two and negative two are the same so now that we've changed subtraction to an addition problem we're going to follow the same rules we did when we were adding two negative integers together since these have the same signs they're both negative we're going to add them together five plus two which is seven and then since they're both negative our result is negative you here we're asked to simplify a pair of polynomial expressions now the keys here are going to be to group like terms and to carry through the signs when adding or subtracting in the first example we have x squared plus 4x plus 2 plus x squared minus 3x minus 4 so we're going to group our like terms together we have from the first expression x squared plus from the second expression x squared plus 4x plus negative 3x so we'll have minus 3x plus 2 plus negative 4 so minus 4 now we already have our like terms group so we're going to add each of these separately x squared plus x squared is 2x squared plus 4x minus 3 X is going to be plus X and plus 2 minus 4 is going to be minus 2 so this is the simplified form of that expression in the second example we have a minus sign here so we have to be careful to carry this minus sign through when we try to include the second and third terms of the second expression so we'll start here we have an x squared and then from the second expression minus x squared plus 3x minus negative x so that's a plus X minus 5 minus plus 3 so minus 3 and once again we have our like terms already grouped so we'll go ahead and add them separately x squared minus x squared just going to be 0 3x plus X is going to be 4x and minus 5 minus 3 is going to be a - eight so we have here this zero can just go away we have 4x minus 8 or 4 times X minus 2 greatest common factor the greatest common factor or GCF of two numbers is the largest value that divides into each of the numbers let's take for instance 18 and 30 you can find the GCF of two numbers by finding the prime factorization of your numbers and then finding what they have in common so the prime factorization of 18 18 is even so it's divisible by 2 so 2 times 9 would be 18 9 is not prime though so we need to break 9 down and the factors of 9 are 3 times 3 so the prime factorization of 18 would be 2 times 3 times 3 3 times 3 is 9 times 2 is 18 I always like to check after I factor to make sure I've got it the prime factorization of 30 30 is divisible by 2 also since it's even so 2 times and 2 times 15 would be 30 but 15 is not prime so we have to break it down into its prime factors and 15s prime factors are 3 times 5 so the prime factorization of 30 is 2 times 3 times 5 2 times 3 is 6 and 6 times 5 is 30 now again to find the GCF we're going to look for what they have in common 18 and 30 have 2 and 3 in common as prime factors so we take those factors they have in common and multiply them 2 times 3 is 6 so the GCF of 18 and 30 is 6 6 is the largest factor that these numbers have in common least common multiple the least common multiple or LCM of two numbers is the smallest value divisible by both the numbers or it's the smallest number that both of your numbers divide into evenly the LCM can be found by finding the GCF or greatest common factors and the remaining factors the LCM is the product of the GCF and those remaining factors for example if we want to define the LCM of 18 and 30 we would first need to factor them 18 is 2 times 9 since 9 is not prime we would factor 9 - 3 times 3 30 is also even so we'll start with 2 30 is 2 times 15 15 is not prime so we factor 15 15 is 3 times 5 the GCF or greatest common factor is 2 times 3 or 6 so our least common multiple is the product of our GCF 6 and the remaining factors the remaining factors are 3 and 5 so 6 times 3 times 5 6 times 5 is 30 30 times 3 is 90 so the smallest number that 18 and 30 will both divide into evenly is 90 mean median and mode are mathematical terms that you should be familiar with the first one is mean now I wrote it up here on the board as arithmetic mean because you oftentimes see it written that way arithmetic mean is just another way of saying average so don't over complicate this define the mean we're going to add the numbers together and then divide them by five we're going to divide them by five because that's how many numbers make up our top numbers see we have one two three four five numbers on top so we're going to divide the number by five you can probably do this calculation in your head two plus two is four plus three is seven plus another seven is 14 plus eleven is 25 so we're going to put 25 over five and divide that out to find that five is the arithmetic mean of this string of numbers the median is the middle number in a string of numbers so for simplification purposes we're going to call a string of numbers like this we're just going to refer to it as a string so in this string you have five numbers the numbers 2 2 3 7 and 11 since there's five numbers here the median or middle number is going to be the third number in the row so we see one two three see there's two numbers to the right of three and two numbers to the left of three so three is in the middle it's the median now when you have six numbers in a string or any amount of even numbers in a string it makes it more complicated so here we have 2 2 3 5 7 and 11 it's hard to find the exact middle number so you 3 and 5 or both in the middle but we can't we can't have two mediums so we have to find the arithmetic mean of 3 and 5 so we're going to add 3 plus 5 then we're going to divide by 2 because we have two numbers here and so the answer is going to be 4 so the median in this case is 4 even though that number doesn't appear in that string another term is mowed that's the most frequently occurring number in a string so here we have the numbers 2 2 3 7 and 11 3 7 and 11 all only appear once whereas 2 appears twice so 2 is going to be our mode in this row of numbers we have 2 2 3 7 seven seven and eleven so over here two was the mode and two appears twice here but here seven appears four times which is more than how many times two appears so in this case seven is going to be the mode these different mathematical terms have different uses to find finding the arithmetic mean can be helpful in getting an overall picture of how much something occurs so take for example trying to find that average income in France you would add all the incomes of people in France together and then divide them by the number of people and then you'd have the arithmetic mean of the incomes in France and you would have an idea of what most people in France make to find the median helps you find the middle of a row of numbers and then finding mode helps you understand what occurs most frequently what number occurs most frequently and then you can ask mode can also be applied to other things like if we had letters we can do a plus a plus B plus C and a would be the mode a is the most commonly occurring letter so that's a quick look at mean median and mode remember that arithmetic mean is basically an average median is the middle number in mode is the most frequently occurring number multiplying and dividing integers when multiplying and dividing integers perform the operation as if both integers were positive and then change the sign of the answer as follows if both the integers have the same sign then their product is positive if the two integers have different signs then the product is negative let's look at some examples on the left side we have four different sets of numbers and in each set the integers have the same sign here we have a positive number times a positive number since they have the same sign our result is positive again we have a set of numbers with the same sign this time they're both negative the result is still positive negative 3 times negative 5 is positive 15 so whether both signs are positive or both signs are negative the result is always positive the same is true with division if both integers are positive the result is positive and if both integers are negative the result is positive so as long as both your integers have the same sign when you're multiplying or dividing then your result is positive this set of numbers each set has different signs each number in the set has different signs here we have a positive 2 times a negative 6 when multiplying one positive number times one negative number the result is always negative and it's the same even if we reverse which number has the negative on it so now when we do negative two times positive 6 we're still doing one negative number times one positive number so that result will still be negative the same is true with division if you're dividing two integers and they have different signs then your result is negative and it doesn't matter if the numerator is negative or the denominator is negative here we have negative ten divided by five so that the negatives are reversed but the result is still negative supplying radicals you can multiply and divide radicals even if they aren't alike to multiply radicals first multiply the coefficients and then do the same with the radicands finally simplify if possible let's look at two examples first we have 3 square root of 5 times 2 square roots of 11 we're going to start by multiplying our coefficients 3 times 2 and then we multiply our radicand so that's the square root of 5 times 11 3 times 2 is 6 square roots of 5 times 11 is 55 55 is not a perfect square and it also doesn't have any perfect square factors we can see because we've pretty much already factored treated here 55 is 5 times 11 and neither one of those are perfect squares either so this can't be simplified therefore 6 square roots of 55 is our answer let's look at 4 square roots of 6 times the square root of 3 we're going to start by multiplying our coefficients 4 times and the coefficient of the square root of 3 is just 1 so 4 times 1 is 4 then we have the square root of 6 times the square root of 3 so we have 4 square roots of 6 times 3 is 18 now 18 is not a perfect square but it does have a perfect square factor so we're going to simplify it for square roots of the perfect square factor of 18 is 9 so I'm going to rewrite it as 9 times 2 which means this is 4 times the square root of 9 times the square root of 2 so we have 4 time the square root of 9 and the square root of 9 is 3 so this is really 4 times 3 times the square root of 2 so when we multiply this all together 4 times 3 is 12 times the square root of 2 12 square roots of 2 you prime factorization of a number is a number written as a multiple of prime numbers again a prime number is a number that's only divisible by 1 in itself meaning it only has factors of 1 and itself let's look at some examples of prime factorization now what two numbers we can come up with that equals 30 when we multiply them 6 & 5 now 5 is a prime number because it's only divisible by 1 and itself 5 but 6 is a composite number which means there are other numbers other than 6 and 1 when you multiply them they will equal 6 you also have 3 & 2 3 is a prime number and 2 is a prime number so when you multiply 3 times 2 times 5 you come up with 30 3 times 2 is 6 times 5 equals 30 let's prime factorization 40 as well again two numbers when you multiply them they equal 44 and 10 10 times 4 equals 44 and 10 or composite numbers they're not prime numbers 4 & 10 or composite numbers while their composite numbers because they're not only divisible by 1 and itself they're also divisible by other numbers so we look at 4 4 is obviously divisible by 1 it's divisible by 4 but it's also divisible by 2 because 2 times 2 can equal will equal 4 then 1010 is a composite number because other than 1 and itself 10 it has two other numbers 5 & 2 when multiplied equals 10 so 10 5 & 2 now 2 is a prime number it's only divisible by 1 and itself 2 as we sit over there with number 35 is a prime number now when we multiply these prime numbers 2 times 2 times 5 times 2 we come up with 40 so 2 times 2 is 4 times 5 times 2 then 4 times 5 is 20 times 2 and then 20 times 2 is equal to 40 this is how we do prime factorization number a prime number is a natural number higher than one that cannot be divided by any other natural number besides one and itself here is a list of the first seven prime numbers two can only be divided by itself and one three can only be divided evenly by itself and one five the same seven so on and so forth notice that in our list of prime numbers the number one is missing one is not considered a prime number since it doesn't have two divisors or two factors one can only be divided by itself one so to be a prime number you have to be able to divide it by two numbers two and only two like on two we can divide it by two and by one but nothing else will divide two evenly notice also we're missing the number four four is not a prime number because it can be divided by one two and four so it can't only be divided by itself and one that's the same with six eight nine ten etc those kinds of numbers are called composite numbers so prime numbers are numbers that can only be divided by 1 and itself and composite numbers would be all the other numbers however one is neither one is neither prime nor composite it's special portions a proportion is a ratio equal to a ratio or a fraction equal to a fraction you've actually been using proportions for a long time anytime you simplify a fraction like ten sixteenths is simplified to five eighths then you've written a proportion because you've written a ratio equal to a ratio now in your proportion you can do what's called the cross products to tell if your ratios are in fact equal for instance with this one ten times eight should equal 16 times five if these ratios are in fact equal ten times eight is 80 and 16 times 5 is 80 so since the cross products are equal that means that our ratios are in fact equal we can also use this cross products principle to solve proportions for instance if instead of 10 16 equals five eighths we had ten sixteenths equals x over eight as in we didn't know what our numerator was then we could use cross products to solve for our variable X so we would cross multiply 16 times X is 16 x equals ten times eight is 80 and then we would solve for X this is 16 times X so the opposite of multiplying would be to divide both sides by 16 then we have 16 divided by 16 which is 1 1 times X is X and 80 divided by 16 is 5 so by using the proportion and the cross products we can find missing values in our proportion like that X is 5 which we already knew from our original proportion rates and unit rates ratios are considered rates when they compare two different units like miles per hour or cost per ounce a unit rate is one in which the numerator of the fraction is compared to a denominator of one unit that way you can tell for instance like how much just one ounce of something costs and you'll see unit rates a lot at the grocery store under or next to the price of an item it'll tell you how much that item costs for every one ounce or for every one thing in the package so let's look at a problem dealing with rates and unit rates Dave is driving 240 miles to his aunt and uncle's house if he gets there in four hours how many and here's the key right here in the question how many miles per hour did he drive on average in that question they're telling you how to set up the problem they're telling you to put miles over hours so we're going to start with that miles per hour and now we can substitute the information in from the problem so they told us how many miles he's going 240 miles and they told us how many hours four hours so we can put that into our rate 240 miles for his four hours so right now this would be considered a rate since we have two different units or miles an hour hours but to determine our unit rate or to figure out his average we would need to divide or simplify our rate to find our unit rate so if we want to have a denominator of one then we're going to have to divide 4 by 4 to get our 1 and if we divide our denominator by 4 then we must so divide our numerator by 4 240 divided by 4 is 60 so again this is miles per hour so what this unit rate tells us is that on average he was going 60 miles for every one hour so your answer could be written as 60 miles per hour is how fast Dave was going on average in math we have rational numbers and we have irrational numbers so we're taking a look at rational numbers which have four different categories and those are integers percents fractions and decimals so we're going to define each of these if you're already somewhat familiar with them we're going to go ahead and define them and give some examples so starting out with integers integer is a positive or negative whole number or zero so nine would be a whole number it would be an integer because it's a positive whole number negative six is an integer because it's a negative whole number then of course zero is an integer because we said it was and so now we move on to percents which it kind of a more complicated way to say percent is a part per 100 so an example of a percent is twenty percent which we could also write is 20 parts per hundred the important thing here is just you know what a percent is like 20 percent in fractions which is a proportion so an example would be 20 divided by 10 or 2 divided by 4 and of course we have decimals so this is where we have an integer maybe one maybe eighty nine maybe zero and we have a decimal place and then we have numbers after the decimal like 20 point five three so we have an integer we have a decimal and then we have numbers after the decimal so integers percent fractions and decimals are all rational numbers usually there are some rules so in order to be a rational number just summarize either has to be an integer like we talked about if it's a number like nine negative six zero it's always going to be a rational number it's more complicated when we come to fractions and decimals if it's a fraction it has to be an integer / an integer and this integer cannot equal zero so it's an integer divided by an integer so seven divided by four cuts it that works now decimal is a little bit more complicated because either has to be a terminating decimal or a repeating decimal so an example of a terminating decimal would be something like 2/4 from that you get 0.5 it ends right there at the 5 or a repeating decimal do 1 divided by 3 and you get zero point 3 3 3 3 it goes on forever but notice we're getting the same number every time so really you could erase all these threes and just put a line over the 3 showing us a repeating decimal what you don't want is a decimal like pi where you have 3.14159265 3 5 9 it just keeps going on and on so there's no ending to it so it's not a terminating decimal and there's no pattern to it so it's not a repeating decimal that's what an irrational number is one that just keeps going like this these numbers they have basically a stopping point and so that's what we're looking for and so the reason I say that a fraction has to be an integer over an integer is because when you divide an integer by an integer you get a clean number you get a decimal that's repeating or terminating when you start dividing a decimal or a fraction by a number or a fraction by another fraction that's when you're going to start getting messy so really if you're wondering whether a fraction is a rational number what you can do is go ahead and divide it out and then look at the decimal and decide whether it's terminating repeating or not ending and that's how you can determine if it was a rational number ratios a ratio is a comparison between two quantities you can think of a ratio like a fraction and since it's just like a fraction a comparison of two numbers it is treated a lot like a fraction meaning you're going to simplify it anytime you can let's look at this problem suppose a sampling of fish from a local pond contains 12 bass 8 catfish 38 minnows and 4 trout what is the ratio so the comparison of two numbers of bass to minutes now when they ask you this question they're telling you exactly how to set your ratio up since it says bass to minnows we're left to right we also read from top to bottom since a ratio is a fraction then they're telling you what your numerator should be and what your denominator should be so our ratio is going to be fast to minnows and now we can substitute how many bass and minnows we have according to our problem we have 12 bass and we have 38 minutes so that ratio would be 12 to 38 but again since this is a fraction we're going to simplify it if we can and 12 and 38 do have a GCF of 2 so we're going to divide 12 by 2 to get 6 and we'll divide 38 by 2 to get 19 therefore our ratio of bass to minnows is 6 to 19 but that's not the only way we can write a ratio we could also write it as 6 to 19 for as 6 to 19 and these all mean the same thing they all give you the same information that for every sick baths in the pond there are 19 minnows in your pond it is important though the order of your numbers since the ratio they were asking for was bass to minnows your fast number should be first and then your minnows number so make sure you put it in the right order that is very important scientific notation scientific notation is a way to express very large or very small numbers in the form a times 10 to the V where a is a number between 1 and 10 and B is an integer we're going to focus on this last part 10 to the B or 10 raised to an integer first so let's look at 10 to the first power 10 to the first power is 10 times itself one time or just 10 10 squared is 10 times itself two times or 110 cubed is 10 times itself three times or 1010 to the fourth power is 10 times itself four times or 10,000 notice as our exponent increases so does our place value of the result as you increase your exponent it's really like you're moving your decimal place one place to the right let's look and see what happens when we raise 10 to a negative exponent to simplify a negative exponent we're going to take the reciprocal so we're going to flip it of the number containing a positive exponent so the reciprocal would be 1 divided by 10 to the first or just 1/10 and 1/10 as a decimal is 1/10 we'll do the same for 10 to the negative 2 so we take the reciprocal with the positive exponent 1 divided by 10 squared which is 1 divided by 10 squared is 10 times 10 which is 100 or 1/100 as a decimal 10 to the negative 3 is 1 divided by 10 to the 3rd power which is 1 divided by 1000 or 1 1000 and then finally 10 to the negative 4 1 divided by 10 to the fourth which is 1 divided by 10,000 or 110 thousand and so we looked at this because with scientific notation we would take a number like 3 times 10 to the second power and what 3 times 10 to the second power is is really 3 times 10 squared which is 100 3 times 100 is 300 or what you can do is since its times 10 to the second power you can take the decimal on 3 and just move it two places to the right since we saw that's what happened with our tens raised to the different exponents we just move the decimal one place to the right as our exponent increased so we can take our decimal 1/3 which if you can't find one it's at the end of the number and move it two places to the right to get the answer three hundred three times 10 to the second power would be three hundred let's look at a number raised to a negative exponent like 4 times 10 to the negative third power so here when we had 10 to the negative first 10 to the negative second etc notice that our our as our exponents were decreasing also our values were decreasing by one by one decimal place each time our numbers got smaller and smaller and smaller our decimal was moved one place to the left each time so we're going to do the same thing with 4 times 10 to the negative third it would be four times and 10 to the negative third we saw was this one 1000 so it's like 4 times 1 1000 which is 4 times 1 1000 which is for thousands and the other way to do this is just to take that to the negative 3 take that power and this negative third power just means to move your decimal three places to the left so we would take our 4 and our decimal is behind it and just move it 1 2 3 places to the left which is the same result we got by doing it the other way numbers and their classifications first we have integers integers are the set of positive and negative numbers including zero which is neither positive nor negative but integers do not include numbers like fractions or decimals that means no mixed numbers either so integers would only be numbers like negative 100 or positive 7 or negative 15 or positive 2 just no fractions no decimals but we can have positive and negative numbers and it also includes zero thanks we have even numbers an even number is any integer so remember by saying it's an integer we're already and not including fractions and decimals so it's any integer no fractions or decimals that can be divided by two without leaving a remainder which means we can divide it evenly so an example of an even number would be 2 4 6 8 10 12 14 16 etc etc etc we could go on forever naming even numbers next we have odd numbers an odd number is again any integer so no fractions or decimals only positive and negative numbers including zero that cannot be divided evenly by two so the exact opposite of an even number even numbers can be divided by two evenly and odd numbers cannot be divided evenly by - and so an example of those numbers would be numbers like 3 5 7 9 11 13 15 17 19 etcetera etc etc a decimal number is a number that uses a decimal point to show the part of the number that is less than 1 for example in the number one and 256 thousandths we use this decimal point to show this part that is less than one rather than using a fraction and so then we get to decimal point a decimal point is a symbol used to separate the ones place from the tenths place like we did in this number right above us this decimal point separates our ones place from our tenths place so it shows you which part of the number is one or greater and which part of the number is less than one finally decimal place that's the position of a number to the right of the decimal point so in this number zero and 123 thousandths the 1 is in the first position to the right of the decimal and so that's the tenths place the two is in the second place to the right of the decimal that's the hundredths place and the 3 is in the third position to the right of the decimal that's the thousandths place and it continues on like that tenths hundredths thousandths X would be ten-thousandths hundred-thousandths and and so on and so forth notice that they all end in a ths so it's not ten thousand it's ten thousandth because it's decimals and so what our whole system is based on is the base 10 system we have ten different digits to work with zero one two three four five six seven eight and nine and we can arrange those numbers in lots of different ways that mean lots of different things but of course there are other systems besides the base 10 system like the binary system and that's what computers use and the binary system that prefix bi means two so in the binary system they only use two numbers 0 and 1 and they put those numbers together to mean different things as well it's thought that maybe we use the base 10 system because we have 10 fingers and maybe 10 toes so it was just easier to come up with something like the base 10 system but these are our numbers and what these different words mean the polar coordinate system is based on a circular graph rather than the square grid of the Cartesian system so I'm sure you're familiar with the rectangular coordinates and those lie on the square grids of the Cartesian plane but the polar coordinate system is different because it's based on a circular graph and so we would write rectangular format like this x and y but we write the polar coordinate system in this format R and then theta that's what we call that symbol right there so R is the distance from the origin the origin is the center of the coordinate plane where the x axis crosses the y axis and so we can call that the radius and so that's easy to remember because radius starts with R and we're looking for variable R now theta is the smallest positive angle in the counterclockwise direction made with the positive horizontal axis that's a positive symbol right there now if we have rectangular format in polar form we have to have a way to convert between the two and so to do that to find polar format we separate it into two equations so first we look for R which is the square root of x squared plus y squared and then theta is the arctangent of Y divided by X when X does not equal 0 now occasionally X is going to equal 0 so we have some special rules for that scenario when x equals 0 and y equals 0 then theta equals 0 when x equals 0 and y is less than 0 then theta equals pi divided by 2 and finally when x equals 0 and y is less than 0 then theta equals 3 pi divided by 2 now if X is positive use the positive square root value for R and if X is negative then use the negative square root value for R now we have to be able to convert in the other direction to so if we have polar format we have to have a way to convert to rectangular format so again we're going to do this by separating it into two separate equations so first we're looking for X we're going to find that by our cosine of theta and then we'll find y by the R sine of theta so that's a look at the rectangular and polar coordinate system

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Id: UU2cVLf4xRU

Length: 74min 19sec (4459 seconds)

Published: Mon Jun 20 2016