Fundamental Concepts of Algebra

Man... Ehhhhh, it's pretty sigma but not like that, nonononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononon 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 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 never-before-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 quirky 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 times 2. It is the same as x plus x, but not x times x. That is x squared. 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 times x, x squared. 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 2. 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 equals mx plus c, or b if you're feeling extra quirky. Dude, you're not even talking yet! This equation and every other 2D graphable 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 set. For every possible x value. I mean that's great for stuff like y equals x, but what about these other things? M stands for Minecraft. Depends on the w-what, hey. NO! No it doesn't. M stands for the slope, the steepness of the graph. So when we are multiplying x by m, it is ste-It is setting the steepness of the graph. And finally for the b, c, c an enemy, an enemy, an enemy, an enemy C is the y-intercept, the y-coordinate where the y-axis and the line meet. 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. Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Mass 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 expression like x, you would add the known values within the brackets and then expand it by multiplying it by X. But for something like X 5 plus 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 5 X plus X Y. This is known as expanding single brackets. I mean but what about double brackets, triple brackets, tetra brackets, uh, skibbity-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 plus 1 x plus 2, you multiply the first by the other first x times x, then add that to the first times by the outer x times 2, then add that by the inner times by the other first 1 times x, and finally add that to the outer times by the other outer 1 times 2. And you get something like this. Now this, remarkably, preposterously, absurdly incongruously is crap. It's crap because it ain't simple. 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? Won't help. But doing brilliant for a couple minutes will. 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. Wanna know how your passwords and home security work so you can protect yourself from Vsauce? How technology works is the cause 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 forward slash find why. You'll also get 20% off an annual premium subscription. Returning to the expression x squared plus 2x plus 1x plus 2, we simplify it down to x squared plus 3x plus 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 uh, let's do that. When you are dealing with powers and roots, there is the coefficient, which you always deal with 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 powers. So in the end, you get 12x to the power 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 in-equal? An inequality or an equation? 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 ol'PAL number line. It'll tell you a range of possible x values in cases like negative 6 is less than x is less than or equal to 2. This is saying that x is a number between negative 6 and 2, including 2. It's important to note that when solving n equations, 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 equations. 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 5 and multiply the bottom by 2. This makes the y coefficient for both equations the same, and thus we just subtract them from each other to get rid of pos. Defenestrate if you will the y variable. We are left with 23x equals 29. Divide both sides by 23 and you get x equals that crap, right then. Launch that back into the simplest original equation, which in case is the top one. Do the math and then solve for y. 2y equals that stuff, therefore y equals that. These are the chords for where the two lines confront each other. But the next part you gotta loglin. Yeah that's like lock in, but like loglin natural log. Enough 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 equals x, this is saying that 2 to the power of x equals 8, therefore x equals 3. When you just see log of x equals y, it is saying that 10 to the power of y equals x. You see, the 10 drank an Invispot and is just cool like that. They are useful. Imagine you had 2 to the power of x equals 16. Now because you guys are big brain, you'll we see that x equals 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 to the power of x equals 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 times log of 2 equals log of 16, we can divide both sides by log of 2 to make x the subject. x equals log of 16 divided by log of 2. This is the same as log of 2 to the power of 4 divided by log of 2. And if you apply the power rule once again, we have 4 times log of 2 divided by log of 2. Now, anything divided by itself that is non-zero will equal 1. So this is saying x equals 4 times 1. In other words, x is 4. Yeah, okay, cool. But what about natural logs? The natural log of x equals y only if e to the power of y equals x, e. Like, random Lord Farquaad Markiplier meme from 20- 2018 yes, that's the one that mathematicians name the number after. And it's roughly equal to 2.718. The natural log of 1 equals 0 and the natural log of e equals 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 times y equals the natural log of x plus the natural log of y and same goes for the opposite, division. So the natural log of x divided by y equals the natural log of x minus the natural log of y. Now let's say that the natural log of x equals 3. This is saying that x equals e to the third. So you literally just raise 2.71a to the third 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 gonna be honest, this section is gonna be pretty nonchalant. What the sigma is sigma. No, no, alright. Oh that sounds bad. It is a Greek letter that mathy 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 equals 5 and b equals 1, and we keep i as it is. It produces 1 plus 2 plus 3 plus 4 plus 5 equals 15. Now the special part is that when we change the i on the right to something more goofy r, for instance i squared plus 5, now if we keep the other parameters as they were, 6 plus 9 plus 14 plus 21 plus 30 equals 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 equals 50, you are adding 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10 plus 11 plus 12 plus 13 plus 14 plus 15 plus 16 plus 17 plus 18 plus 19 plus 20 plus 21 plus 22 plus 23 plus 24 plus 25 plus 26 plus 28 plus 29 plus 30 plus 31 plus 32 plus 34 plus 35 plus 36 plus 37 plus 38 plus 39 plus 40 plus 41 plus 32 plus 34 plus 34 plus 35 plus 37 plus 48 plus 49 plus 90 plus 50. Now that would be very painful had this formula not existed. Simply plug 50 into n and we get 1275. Riemann's Law A good ol'jolly ol'chap, Bernard Riemann, was playing some Go Fitch with his dog. And whenever he threw the ball, he noticed it travelled in a curve. I have talked about finding the area under a curve in my calculus for gen z part one and there's gonna be more to come go watch that but this guy took a marginally different approach so imagine you have the curve formed by negative x squared plus one and you're Reeman 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 subintervals. So delta x equals point b minus point a divided by 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 subintervals that we use to... calculate with. Most commonly this is the midpoint. This little fancy thingy here is the midpoint for a given subinterval and it is equal to xi minus one plus xi divided by two where the other xi things are the left and right points of the subinterval. Man this has to be the most amount of yapping I've done on this channel. Uh well you think I'm gonna stop? No of course not. We plug this value along with our original subinterval 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 Under my bed and in fact that thing is so scared of having to add So we resort to the next best thing after Desmos fails to do his job It's probably 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. This video is so bad, actually. How am I-how am I gonna edit this? This is ridiculous.

Which is just painful and not a good way to introduce these numbers. So instead of amputation, we resort 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 never-before-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 quirky 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 times 2. It is the same as x plus x, but not x times x.

That is x squared. 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 times x, x squared.

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 2. 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 equals mx plus c, or b if you're feeling extra quirky.

Dude, you're not even talking yet! This equation and every other 2D graphable 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 set. For every possible x value.

I mean that's great for stuff like y equals x, but what about these other things? M stands for Minecraft. Depends on the w-what, hey.

NO! No it doesn't. M stands for the slope, the steepness of the graph. So when we are multiplying x by m, it is ste-It is setting the steepness of the graph. And finally for the b, c, c an enemy, an enemy, an enemy, an enemy C is the y-intercept, the y-coordinate where the y-axis and the line meet.

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. Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Wars Bed Mass 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 expression like x, you would add the known values within the brackets and then expand it by multiplying it by X. But for something like X 5 plus 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 5 X plus X Y. This is known as expanding single brackets. I mean but what about double brackets, triple brackets, tetra brackets, uh, skibbity-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 plus 1 x plus 2, you multiply the first by the other first x times x, then add that to the first times by the outer x times 2, then add that by the inner times by the other first 1 times x, and finally add that to the outer times by the other outer 1 times 2. And you get something like this. Now this, remarkably, preposterously, absurdly incongruously is crap. It's crap because it ain't simple.

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? Won't help. But doing brilliant for a couple minutes will.

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. Wanna know how your passwords and home security work so you can protect yourself from Vsauce? How technology works is the cause 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 forward slash find why. You'll also get 20% off an annual premium subscription.

Returning to the expression x squared plus 2x plus 1x plus 2, we simplify it down to x squared plus 3x plus 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 uh, let's do that. When you are dealing with powers and roots, there is the coefficient, which you always deal with 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 powers.

So in the end, you get 12x to the power 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 in-equal?

An inequality or an equation? 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 ol'PAL number line. It'll tell you a range of possible x values in cases like negative 6 is less than x is less than or equal to 2. This is saying that x is a number between negative 6 and 2, including 2. It's important to note that when solving n equations, 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 equations. 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 5 and multiply the bottom by 2. This makes the y coefficient for both equations the same, and thus we just subtract them from each other to get rid of pos. Defenestrate if you will the y variable. We are left with 23x equals 29. Divide both sides by 23 and you get x equals that crap, right then.

Launch that back into the simplest original equation, which in case is the top one. Do the math and then solve for y. 2y equals that stuff, therefore y equals that. These are the chords for where the two lines confront each other. But the next part you gotta loglin.

Yeah that's like lock in, but like loglin natural log. Enough 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 equals x, this is saying that 2 to the power of x equals 8, therefore x equals 3. When you just see log of x equals y, it is saying that 10 to the power of y equals x. You see, the 10 drank an Invispot and is just cool like that. They are useful. Imagine you had 2 to the power of x equals 16. Now because you guys are big brain, you'll we see that x equals 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 to the power of x equals 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 times log of 2 equals log of 16, we can divide both sides by log of 2 to make x the subject. x equals log of 16 divided by log of 2. This is the same as log of 2 to the power of 4 divided by log of 2. And if you apply the power rule once again, we have 4 times log of 2 divided by log of 2. Now, anything divided by itself that is non-zero will equal 1. So this is saying x equals 4 times 1. In other words, x is 4. Yeah, okay, cool.

But what about natural logs? The natural log of x equals y only if e to the power of y equals x, e. Like, random Lord Farquaad Markiplier meme from 20- 2018 yes, that's the one that mathematicians name the number after. And it's roughly equal to 2.718.

The natural log of 1 equals 0 and the natural log of e equals 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 times y equals the natural log of x plus the natural log of y and same goes for the opposite, division. So the natural log of x divided by y equals the natural log of x minus the natural log of y. Now let's say that the natural log of x equals 3. This is saying that x equals e to the third.

So you literally just raise 2.71a to the third 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 gonna be honest, this section is gonna be pretty nonchalant. What the sigma is sigma. No, no, alright.

Oh that sounds bad. It is a Greek letter that mathy 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 equals 5 and b equals 1, and we keep i as it is. It produces 1 plus 2 plus 3 plus 4 plus 5 equals 15. Now the special part is that when we change the i on the right to something more goofy r, for instance i squared plus 5, now if we keep the other parameters as they were, 6 plus 9 plus 14 plus 21 plus 30 equals 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 equals 50, you are adding 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10 plus 11 plus 12 plus 13 plus 14 plus 15 plus 16 plus 17 plus 18 plus 19 plus 20 plus 21 plus 22 plus 23 plus 24 plus 25 plus 26 plus 28 plus 29 plus 30 plus 31 plus 32 plus 34 plus 35 plus 36 plus 37 plus 38 plus 39 plus 40 plus 41 plus 32 plus 34 plus 34 plus 35 plus 37 plus 48 plus 49 plus 90 plus 50. Now that would be very painful had this formula not existed. Simply plug 50 into n and we get 1275. Riemann's Law A good ol'jolly ol'chap, Bernard Riemann, was playing some Go Fitch with his dog.

And whenever he threw the ball, he noticed it travelled in a curve. I have talked about finding the area under a curve in my calculus for gen z part one and there's gonna be more to come go watch that but this guy took a marginally different approach so imagine you have the curve formed by negative x squared plus one and you're Reeman 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 subintervals. So delta x equals point b minus point a divided by 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 subintervals that we use to...

calculate with. Most commonly this is the midpoint. This little fancy thingy here is the midpoint for a given subinterval and it is equal to xi minus one plus xi divided by two where the other xi things are the left and right points of the subinterval. Man this has to be the most amount of yapping I've done on this channel. Uh well you think I'm gonna stop?

No of course not. We plug this value along with our original subinterval 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 Under my bed and in fact that thing is so scared of having to add So we resort to the next best thing after Desmos fails to do his job It's probably 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. This video is so bad, actually. How am I-how am I gonna edit this?

This is ridiculous.

Transcript for:Fundamental Concepts of Algebra

Transcript for:
Fundamental Concepts of Algebra