[Applause] man it's pretty Sig but not like that Sigma notation one of the many topics I'm going to cover in this video the entirety of elge not really explained in 15 minutes part one let's go we start by counting humans have five thingies on each hand which makes it easy to count from 1 to 10 but what's not easy is negative numbers negative thingies which is just painful and not a good way to introduce these numbers so instead of amputation we resort to to a line a road that spans infinitely far in the positive numbers and infinitely far in the negatives great you have unlocked the number line so if you were to choose a random number in this set of numbers X then you would most likely get a neverbe seen value that will never be written down ever again in the entire existence of the universe anyway what is X x is a variable an object in which values can be stored algebraic expressions which are basically quy little math statements always have variables when you see something like this it is saying that there are two lots of the value X the coefficient of x is 2 so it's just x * 2 it is the same as x + x but not x * X that is x s but why do we call it squared well imagine X is the side length of a square we know all squares must have equal sides that are perpendicular given that X is one length and all lengths are equal we know that all lengths are X we also know that the area of a rectangle is the length of the perpendicular sides multiplied so the area of a square is x * X x² awesome now you know what a square is but when we look at this what the flip dude what the flip what the flip is the two well it is the exponent a value that makes straight lines go like this straight lines when graphed are lines made by the equation y = mx + C or B if you're feeling extra quy the animatronic characters here do get a bit crunky at night this's equation and every other the 2D graph equation is asking when X is equal to something what is y equal to so when you are graphing it with Desmos the program is finding Y like me find y join the Discord Ser but every possible x value I mean that's great for stuff like yal X but what about these other things m stands for Minecraft depends on the who hey hey no it doesn't uh m stands for the slope the steepness of the graph so when we are multiplying X by m it is it is setting the steepness of the graph and finally for the b c CN n minut enemy spotted counter terrorist win C is the Y intercept the y coordinate with a Y AIS and the line meat so to quickly recap for each possible x coordinate the respective y-coordinate is equal to the steepness multiplied by said X then shifted up or down by the Y intercept bedw Wars bad Mass Bam Dash Pam Dash Pim Dash p Mass p dash BH all stupid acronyms that mean the exact same thing in Expressions the calculation process is brackets comes first then exponents then multiplication then division then addition and subtraction but what does it mean for brackets to come first well it depends if the expression has unknown variables within the brackets or not for an like x 5 + 6 you would add the known values within the brackets then expand it by multiplying it by X but for something like x 5 + y we can't add 5 to Y at the start because we don't know what Y is so we just do the expansion to get 5x plus XY this is known as expanding single brackets I mean but what about double brackets triple brackets Tetra brackets uh scoy toilet brackets in all expansions we are multiplying everything on the outside by everything in the inside one of the easiest methods is a thing called foil first outer inner last for something like x + 1 x + 2 you multiply the first by the other first x * X then add that to the first times by the outer x * 2 then add that by the inner Times by the other first 1 * X and finally add that to the alter Times by the other alter 1 * two and you get something like this now this remarkably preposterously absurdly in congruously is crap it's crap because it ain't simple I mean that's obvious but one of the big parts of basic algebra is the process of simplification it makes your life easier while solving equations because it turns an expression into the most efficient and smallest it can be but you know what else makes your life easier while solving equations today's sponsor brilliant.org pain comes in many forms if you're anything like me getting a well-deserved not achieved and arithmetic is not a great feeling with Brilliance Vector measurement and geometry and predicting what data called I went from being a Doomer to a Doomer with more of a clue than usual that sounds great it is great doing brilliant for even a couple seconds a day won't help but doing brilliant for a couple minutes well problem solving skills are built up through the Hands-On Math and Science lessons which are six times better than watching lecture videos and much better than pure memorization I know that because I can't remember anything I know that because I can't remember anything I know that because I can't remember anything have you ever wanted to know how NPCs work bam multiple courses on AI and large language models want to know how your passwords and home security works so you can protect yourself from Vis how technology works is the cuse for you so if this sounds mildly delectable to try everything brilliant has to offer for free for a full 30 days use my link brilliant.org slind why you'll also get 20% off an annual premium subscription returning to the expression x^2 + 2 x + 1 x + 2 we simplify it down to X2 + + 3x + 2 what we did is we separated each term from each other based on coefficient and exponent then added it together we never properly went over powers and exponents so um let's do that when you're dealing with powers and Roots there is the coefficient which you always dealed first then there's the base and the exponent if the bases are the same we add powers when multiplying bases and subtract when dividing so for that expression as I said you deal with coefficients first 12 then add the so in the end you get 12x to^ of 3 if an equation has identical bases and coefficients on both sides of the equal sign then the powers must be equal this is very helpful in doing basic exponent problems yeah I'm yapping on about stuff that is equal but what if it's inequal an inequality or inequation this means less than this means greater than this means less than or equal to and you guessed it this means greater than or equal to equal to what any number or expression that fits on our good old pal number line it'll tell you a range of possible X values in cases like -6 is less than x is less than or equal to 2 this is saying that X is a number between -6 and 2 including 2 it's important to note that when solving inequations if you divide or multiply both sides by a negative number the sign flips this also happens when taking the reciprocal this all seems to take a lot of time to solve so what about something that happens all at once simultaneously simultaneous equation well actually they still take a lot of steps and it's probably more than the other ones but shut up shut up this is a system of equations a set of equations with more than one unknown variable which we solve for simultaneously by working on both at once there are two techniques for solving these elimination and substitution for this example we multiply the entire top equation by five and multiply the Bottom by two this makes the Y coefficient for both equations the same and thus we just subtract them from each other to get rid of toss def straight if you will the Y variable we are left with 23x = 29 divide both sides by 23 and you get x equals that crap right there launch that back into the simplest original equation which in this case is the top one do the math and then solve for y 2 y equals that stuff therefore y equals that these are the chords for where the two lines confront each other for this next part you got to log Lin it's like log in but like log natural law with the shenanigans we don't have time here logarithms trees that like music they are the opposite to exponents I mean they're not Roots but what they do is they find which power a base has to be raised to to equal a number it is written like log 2 of 8 = x this is saying that 2^ of x = 8 therefore x = 3 when you just see log of x = y it is saying that 10 ^ of y = x you see the 10 drank an invers po and is just cool like that they are useful imagine you had 2 the^ of x = 16 now because you guys are big brain you'll quickly see that x = 4 but let's imagine you have the same IQ as my dog I don't have a dog so you really don't have any IQ first step is to take a log of both sides log of 2 ^ x = log of 16 now this next step involves something called the power rule where the exponent of the number in the brackets can be moved outside of the log and become a coefficient now that you have x * log of 2 = log of 16 we can divide both sides by log of 2 to make X the subject xal log of 16 / log of 2 this is the same as log of 2^ 4 / log of 2 and if you apply the power rule once again we have 4 * log of 2 / log of 2 now anything divided by itself that is non zero will equal 1 so this is saying x = 4 * 1 in other words x is 4 yeah okay cool but what about natural logs the natural log of x = y only if e to^ of y = x e like random Lord farquad Mark apply a meme from 2018 yes that's the one that mathematicians named the number after and it's roughly equal to 2.718 the natural log of 1al 0 and the natural log of e = 1 there is a thing called the product property which applies to both natural and normal logs that pretty much state that the natural log of x * y equal the natural log of x plus the natural log of Y and same goes for the the opposite division so the natural log of x / y equal the natural log of x minus the natural log of Y now let's say that the natural log of x = 3 this is saying that x = e to 3r so you literally just raise 2.718 to the 3r and you get x equals roughly 20 it's not that deep now at the start of this video I did promise to cover Sigma notation and I'm going to be honest this section is going to be pretty nonchalant what the sigma is Sigma what no no all right I know that sounds bad it is a Greek letter that maty boys use as the notation for summation the total you get from adding together all numbers in an iterative sequence there are three inputs the thingy at the top a the thing on the right I and the thing at the bottom B with the current conditions this expression is saying that as I increases from B to a counting up one at a time add the numbers together so let's imagine that a = 5 and Bal 1 and we keep I as it is it produces 1 + 2 + 3 + 4 + 5 = 15 now the special part is that when we change the I on the right to something more goofy R for instance i^ 2 + 5 now if we keep the other parameters as they were 6 + 9 + 14 + 21 + 30 = 80 now there are a couple useful common formulas that revolve around summation this one gives us the sum of all the natural numbers so positive integers from 1 to n so if n = you are adding 1 + 2 + 3 + 4 + 5 + 6+ 7 + 8+ 9+ 10+ 11+ 12 14 15+ 16 17+ 18+ 19 20 plus 50 now that would be very painful had this formula not existed simply plug 50 into n and we get 1,275 a good o Jolly o chap Bernard rean was playing some go Fitch with his dog and whenever he threw the ball he noticed it traveled in a curve I have talked about finding the area under a curve in my calculus for Gen Zed part one there's going to be more to come go watch that but this guy took a marginally different approach so imagine you have the curve formed by x^2 + 1 and you're rayan and you're looking for a way to find the area under this curve between two x values this range is called the interval it is a closed interval so it includes the two numbers themselves and We Begin by dividing the segment of the x-axis into Parts the amount of parts that you divide it into the more accurate the area will be let's say that you divide it into 100 subsections or sub intervals so Delta x equal point B minus point a / n where n is 100 and point B is the greatest number out of the two points now we have to have a value Within These sub intervals that we use to calculate with most commonly this is the midpoint this little fancy thingy here is the midpoint for a given sub interval and it is equal to x i - 1 + x i / 2 where the other XI things are the left and right points of the sub interval man this has to be the most amount of yapping I've done on this channel uh well you think I'm going to stop no of course not we plug this value along with our original sub interval calculator into this sum and what we have is an elegant equation that if we wanted to manually add together would make me cry myself to sleep even with that thingy uh under my bed and in fact that thing is so scared of having to add 100 sub intervals together after substituting them into X x^2 + 1 so we resort to the next best thing after Desmos fails to do its job it's probably it's it's definitely me just being dumb and get GPT to do it for us and what do you know 1.33 recurring is the area and when we plug the function into a definite integral we get the same answer but find why my beloved where is the abstract algebra the linear algebra the real analysis where is my vertex operator algebra whatever the hell that is okay I lied this video does not cover all algebra but but I do want to cover at least some of it so if you guys actually found this video remotely useful and if it gets more than three views I'll consider upping my game for the next one but until then catch you later thanks for watching watching this video is so bad like actually like how how am I going to edit this this is ridiculous so bad I won feel much watch this [Music]