WEBVTT 00:00:00.700 --> 00:00:05.360 Welcome to algebra. I know what some of you are thinking: why 00:00:05.360 --> 00:00:09.400 are there letters in my math? What did the alphabet ever do 00:00:09.400 --> 00:00:12.940 to deserve this? Here is the answer — and by the end 00:00:12.940 --> 00:00:17.620 of today it will genuinely make sense. Algebra is not arithmetic wearing 00:00:17.620 --> 00:00:23.040 a disguise. It is a different kind of thinking altogether. Arithmetic asks: 00:00:23.040 --> 00:00:27.800 what is three times four? Algebra asks: what number, when tripled and 00:00:27.800 --> 00:00:34.000 increased by five, gives twenty-six? One is computing. The other is reasoning. 00:00:34.000 --> 00:00:38.380 That shift — from computing a specific answer to reasoning about what 00:00:38.380 --> 00:00:42.080 an unknown must be — is one of the most powerful ideas 00:00:42.080 --> 00:00:46.040 humans ever discovered. It is also about twelve hundred years old, and 00:00:46.040 --> 00:00:49.520 it was invented to solve inheritance disputes in Baghdad. Let us go 00:00:49.520 --> 00:00:50.500 find out how. 00:00:51.700 --> 00:00:57.460 Here is the fundamental problem that algebra was invented to solve. Arithmetic 00:00:57.460 --> 00:01:03.240 is brilliant at computing. Tell it that three times four equals twelve, 00:01:03.240 --> 00:01:06.600 it handles that effortlessly. But what happens when you do not know 00:01:06.600 --> 00:01:11.200 one of the numbers? A merchant earns three gold coins for every 00:01:11.200 --> 00:01:14.440 sheep he sells, and by end of day he has earned twenty-one 00:01:14.440 --> 00:01:19.380 gold coins. How many sheep did he sell? You could guess and 00:01:19.380 --> 00:01:25.320 check — try seven, which gives twenty-one, done. But what if the 00:01:25.320 --> 00:01:31.040 numbers were messier? What if the problem involved two unknown quantities at 00:01:31.040 --> 00:01:39.380 once? Guess-and-check breaks down. You need a method that always works, no 00:01:39.380 --> 00:01:44.740 matter how complicated the numbers get. That is what algebra provides — 00:01:44.740 --> 00:01:49.400 a systematic way of reasoning about unknowns, turning the question from 'can 00:01:49.400 --> 00:01:53.400 I guess the answer?' to 'I can prove what the answer must 00:01:53.400 --> 00:01:56.548 be.' That shift is enormous. 00:01:57.767 --> 00:02:02.447 Let us clear up the single biggest confusion about algebra right now. 00:02:02.447 --> 00:02:07.547 The letter x is not a mysterious unknown that could be anything. 00:02:07.547 --> 00:02:11.067 It is a placeholder — a label for a specific number that 00:02:11.067 --> 00:02:15.327 we are going to find. When you see 2x plus 3 equals 00:02:15.327 --> 00:02:19.987 11, that equation is making a very specific claim: there is a 00:02:19.987 --> 00:02:24.287 number, and if you double it and add three, you get eleven. 00:02:24.287 --> 00:02:28.427 Your job is to find that number. The symbol x just marks 00:02:28.427 --> 00:02:33.047 the spot where that number goes. You could use any letter — 00:02:33.047 --> 00:02:38.247 mathematicians sometimes use n, or a, or even a smiley face. The 00:02:38.247 --> 00:02:43.007 letter does not matter. What matters is the logic of the equation. 00:02:43.007 --> 00:02:46.587 And here is the key insight: once you find x equals four, 00:02:46.587 --> 00:02:51.007 you can check it. Plug four back in: two times four is 00:02:51.007 --> 00:02:56.807 eight, plus three is eleven. The equation is satisfied. That checkability is 00:02:56.807 --> 00:03:01.271 what makes algebra trustworthy — you can always verify your answer. 00:03:02.500 --> 00:03:05.420 The word algebra is about twelve hundred years old, and it is 00:03:05.420 --> 00:03:11.700 Arabic. In 820 CE, a mathematician named Muhammad ibn Musa al-Khwarizmi was 00:03:11.700 --> 00:03:14.880 working in Baghdad at the House of Wisdom, a kind of ancient 00:03:14.880 --> 00:03:19.340 research university funded by the Caliph. He wrote a book whose title 00:03:19.340 --> 00:03:25.080 translates roughly to 'The Compendious Book on Calculation by Completion and Balancing.' 00:03:25.080 --> 00:03:28.620 The Arabic phrase in the title — al-jabr, meaning the reunion of 00:03:28.620 --> 00:03:32.580 broken parts or the completion of an equation — became, in Latin 00:03:32.580 --> 00:03:38.020 translation, the word algebra. Al-Khwarizmi did not write his book for fun. 00:03:38.020 --> 00:03:42.540 He wrote it to help judges handle inheritance cases. When someone died, 00:03:42.540 --> 00:03:46.520 the property had to be split among relatives according to Islamic law, 00:03:46.520 --> 00:03:51.100 and those calculations could get complicated fast. Algebra was a practical tool 00:03:51.100 --> 00:03:55.740 for practical problems. Every time you write an equation today, you are 00:03:55.740 --> 00:03:59.520 using a method that has been trusted by judges, merchants, and engineers 00:03:59.520 --> 00:04:04.260 for over a thousand years. And here is the bonus trivia: the 00:04:04.260 --> 00:04:09.540 word algorithm comes from the Latin version of al-Khwarizmi's own name. His 00:04:09.540 --> 00:04:13.220 methods were so systematic that his name became the word for a 00:04:13.220 --> 00:04:14.980 step-by-step procedure. 00:04:16.200 --> 00:04:19.740 Here is the mental image that will make algebra click for the 00:04:19.740 --> 00:04:25.100 rest of your life. Picture a perfectly balanced scale — the old-fashioned 00:04:25.100 --> 00:04:29.740 kind, with two pans hanging from a bar. Something is sitting on 00:04:29.740 --> 00:04:33.840 the left pan. Something is sitting on the right pan. They weigh 00:04:33.840 --> 00:04:38.380 exactly the same. Now: if you add a weight to the left 00:04:38.380 --> 00:04:42.300 pan, the scale tips left. But if you add the same weight 00:04:42.300 --> 00:04:47.720 to both pans, the scale stays balanced. That is an equation. The 00:04:47.720 --> 00:04:51.320 left side and right side are equal. The equals sign is the 00:04:51.320 --> 00:04:55.080 pivot of the balance. Anything you do to one side of an 00:04:55.080 --> 00:04:59.220 equation, you must do to the other, or you destroy the balance 00:04:59.220 --> 00:05:05.080 and the equation becomes false. Add three to both sides: still balanced. 00:05:05.080 --> 00:05:10.800 Subtract seven from both sides: still balanced. Multiply everything by two: still 00:05:10.800 --> 00:05:15.160 balanced. This is the one rule of algebra. It is not a 00:05:15.160 --> 00:05:19.340 trick or a shortcut — it is a logical law. And it 00:05:19.340 --> 00:05:23.940 is why every step you take when solving an equation is reversible: 00:05:23.940 --> 00:05:26.448 you can always go back and check your work. 00:05:27.667 --> 00:05:33.187 Let us dissect the equation two x plus three equals eleven, piece 00:05:33.187 --> 00:05:37.827 by piece, so nothing looks mysterious again. The two in front of 00:05:37.827 --> 00:05:42.027 x is called the coefficient — it is the number that multiplies 00:05:42.027 --> 00:05:46.887 the variable. It tells you: take x, and double it. The x 00:05:46.887 --> 00:05:50.587 itself is the variable — the placeholder for the number we are 00:05:50.587 --> 00:05:57.207 hunting. Together, two x is a term: coefficient times variable, traveling as 00:05:57.207 --> 00:06:01.147 a unit. The plus sign connects our terms. The three is a 00:06:01.147 --> 00:06:05.787 constant — a plain number with no variable attached. It just sits 00:06:05.787 --> 00:06:10.287 there, adding three to whatever two x is. The equals sign is 00:06:10.287 --> 00:06:13.487 the pivot of the whole thing: it is claiming that the expression 00:06:13.487 --> 00:06:17.927 on the left, whatever it evaluates to, is exactly the same as 00:06:17.927 --> 00:06:23.367 eleven on the right. Now solving: we want x alone. Subtract three 00:06:23.367 --> 00:06:27.827 from both sides — two x equals eight. Divide both sides by 00:06:27.827 --> 00:06:33.267 two — x equals four. Check: two times four plus three is 00:06:33.267 --> 00:06:39.067 eleven. Correct. Every equation you will ever see in algebra is built 00:06:39.067 --> 00:06:41.059 from these same ingredients. 00:06:42.267 --> 00:06:46.467 Here is something that might surprise you: all of algebra ultimately rests 00:06:46.467 --> 00:06:50.387 on just a handful of properties — rules about how numbers behave 00:06:50.387 --> 00:06:53.667 that have been true for as long as numbers have existed. The 00:06:53.667 --> 00:06:58.307 commutative property says that order does not matter for addition and multiplication: 00:06:58.307 --> 00:07:02.347 three plus five is the same as five plus three. The associative 00:07:02.347 --> 00:07:06.427 property says that grouping does not matter: two plus three plus four 00:07:06.427 --> 00:07:10.567 equals two plus seven equals nine, no matter how you bracket it. 00:07:10.567 --> 00:07:14.727 The distributive property is the one you will use the most: multiplying 00:07:14.727 --> 00:07:18.687 a number by a sum is the same as multiplying that number 00:07:18.687 --> 00:07:23.007 by each part of the sum separately. So three times the quantity 00:07:23.007 --> 00:07:27.867 five plus two equals three times five plus three times two, which 00:07:27.867 --> 00:07:33.007 is fifteen plus six, which is twenty-one. These are not arbitrary rules 00:07:33.007 --> 00:07:38.427 someone made up. They are descriptions of how quantities actually behave. The 00:07:38.427 --> 00:07:42.147 reason algebra works so reliably is that it is built on these 00:07:42.147 --> 00:07:48.051 logical foundations — not memorized procedures, but provable truths. 00:07:49.267 --> 00:07:53.707 Let us walk through a complete worked example from start to finish: 00:07:53.707 --> 00:07:57.627 three x minus seven equals fourteen. This is the kind of equation 00:07:57.627 --> 00:08:00.707 that trips people up if they try to do it by feel, 00:08:00.707 --> 00:08:04.567 but it is completely mechanical if you follow the balance rule. Our 00:08:04.567 --> 00:08:08.587 goal is to get x by itself on one side. Step one: 00:08:08.587 --> 00:08:11.667 the minus seven is in the way. To get rid of it, 00:08:11.667 --> 00:08:17.647 we add seven to both sides — because adding seven undoes subtracting 00:08:17.647 --> 00:08:24.287 seven. Left side: three x minus seven plus seven equals three x. 00:08:24.287 --> 00:08:29.527 Right side: fourteen plus seven equals twenty-one. So now we have three 00:08:29.527 --> 00:08:35.287 x equals twenty-one. The scale is still balanced. Step two: x is 00:08:35.287 --> 00:08:39.287 being multiplied by three. To get rid of that, we divide both 00:08:39.287 --> 00:08:45.167 sides by three. Left side: three x divided by three equals x. 00:08:45.167 --> 00:08:52.407 Right side: twenty-one divided by three equals seven. So x equals seven. 00:08:52.407 --> 00:08:57.987 Always check: plug seven back into the original equation. Three times seven 00:08:57.987 --> 00:09:04.447 minus seven equals twenty-one minus seven equals fourteen. Correct. That check is 00:09:04.447 --> 00:09:09.091 not optional — it is the proof that you found the right answer. 00:09:10.300 --> 00:09:14.940 Algebra is extraordinarily powerful, but it has a blind spot: it solves 00:09:14.940 --> 00:09:18.540 whatever equation you give it, without knowing whether that equation was a 00:09:18.540 --> 00:09:22.620 good description of the situation. Imagine you are asked how many students 00:09:22.620 --> 00:09:26.420 can fit in a room, you set up the equation incorrectly, and 00:09:26.420 --> 00:09:32.480 algebra faithfully gives you negative twelve. Negative twelve students. Algebraically, that is 00:09:32.480 --> 00:09:36.700 a perfectly valid answer to the equation you wrote. In reality, it 00:09:36.700 --> 00:09:40.840 is nonsense. The equation was wrong, so the answer is wrong. This 00:09:40.840 --> 00:09:43.600 is not a failure of algebra — it is a reminder that 00:09:43.600 --> 00:09:47.380 the tool only works as well as the model. In practice, this 00:09:47.380 --> 00:09:51.720 means two things. First, always check whether your answer makes sense in 00:09:51.720 --> 00:09:56.000 the original context. Second, learning to set up the right equation is 00:09:56.000 --> 00:09:59.780 just as important as learning to solve it. Plenty of bright students 00:09:59.780 --> 00:10:03.860 can solve equations perfectly and still get the wrong answer, because they 00:10:03.860 --> 00:10:07.960 wrote down the wrong equation in the first place. Modeling reality in 00:10:07.960 --> 00:10:11.100 algebra is a skill all its own, and it is one we 00:10:11.100 --> 00:10:13.156 will practice throughout this course. 00:10:14.367 --> 00:10:18.787 It is worth pausing to appreciate exactly what is different between arithmetic 00:10:18.787 --> 00:10:24.087 and algebra, because they feel similar but they are not. Arithmetic deals 00:10:24.087 --> 00:10:28.367 in specific numbers. You put numbers in, you get a number out. 00:10:28.367 --> 00:10:33.767 Four times seven equals twenty-eight. Full stop. Algebra deals in relationships between 00:10:33.767 --> 00:10:38.547 quantities, some of which may be unknown. Four times x equals twenty-eight 00:10:38.547 --> 00:10:42.287 — now I am describing a relationship, and I want to know 00:10:42.287 --> 00:10:47.287 which value of x makes it true. The payoff is generalization. Consider 00:10:47.287 --> 00:10:51.507 the arithmetic fact that any number times two equals that number plus 00:10:51.507 --> 00:10:56.327 itself. Writing that out in arithmetic would take forever — you would 00:10:56.327 --> 00:11:00.707 have to list every case. In algebra, you write it once: two 00:11:00.707 --> 00:11:07.787 times x equals x plus x. One equation, infinite cases covered. Every 00:11:07.787 --> 00:11:14.367 formula in physics, every model in economics, every algorithm in computer science 00:11:14.367 --> 00:11:18.767 is written in the language of algebra, because algebra is the language 00:11:18.767 --> 00:11:25.007 of general relationships. Arithmetic is useful for specific computations. Algebra is useful 00:11:25.007 --> 00:11:28.191 for thinking about how things relate to each other in principle. 00:11:29.400 --> 00:11:33.960 Every student who has ever learned algebra has made these mistakes. Not 00:11:33.960 --> 00:11:37.740 the students who struggle — every student, including the ones who later 00:11:37.740 --> 00:11:41.860 became mathematicians. So let us name them now, so you can spot 00:11:41.860 --> 00:11:46.200 them when they happen. The biggest one is forgetting that the equals 00:11:46.200 --> 00:11:50.580 sign demands symmetry. Students add three to the left side to clear 00:11:50.580 --> 00:11:54.340 a constant, and forget to add three to the right side. The 00:11:54.340 --> 00:11:58.280 equation is now wrong, but everything after that looks like valid algebra. 00:11:58.280 --> 00:12:02.940 The result is a confident wrong answer — the worst kind. The 00:12:02.940 --> 00:12:07.060 second common mistake is combining unlike terms. Two x and three are 00:12:07.060 --> 00:12:11.220 not five x. Two x means two of whatever x is. Three 00:12:11.220 --> 00:12:14.980 is just three. They cannot be added until you know what x 00:12:14.980 --> 00:12:20.720 is. The third classic mistake is sign errors, especially with negatives. When 00:12:20.720 --> 00:12:25.300 you subtract a negative — when you have, say, five minus negative 00:12:25.300 --> 00:12:31.200 three — the two negatives combine and you get eight, not two. 00:12:31.200 --> 00:12:35.380 Negative numbers are confusing for everyone at first. The solution to all 00:12:35.380 --> 00:12:40.760 three mistakes is the same: slow down, write every step, and always 00:12:40.760 --> 00:12:42.624 check your answer at the end. 00:12:44.233 --> 00:12:49.553 In the professional world, nobody does long algebra by hand when software 00:12:49.553 --> 00:12:55.213 can do it faster and without arithmetic errors. Tools like Desmos, Wolfram 00:12:55.213 --> 00:12:59.313 Alpha, and GeoGebra are free, run in a web browser, and can 00:12:59.313 --> 00:13:05.113 solve equations, graph functions, and check work instantly. Scientists and engineers use 00:13:05.113 --> 00:13:10.913 computer algebra systems — programs like Mathematica or Python's SymPy library — 00:13:10.913 --> 00:13:15.613 to handle expressions that would take hours to manipulate by hand. But 00:13:15.613 --> 00:13:18.593 here is the thing none of those tools can do: they cannot 00:13:18.593 --> 00:13:23.893 read the problem. They cannot decide what the unknown quantity is, what 00:13:23.893 --> 00:13:28.913 equation captures the situation, or whether the answer that came back makes 00:13:28.913 --> 00:13:33.773 physical sense. A computer algebra system will happily tell you that x 00:13:33.773 --> 00:13:39.313 equals negative forty-seven with complete confidence. It has no way of knowing 00:13:39.313 --> 00:13:44.733 that negative forty-seven apples is absurd. Setting up the equation, interpreting the 00:13:44.733 --> 00:13:49.033 answer, and checking whether it fits reality — all of that is 00:13:49.033 --> 00:13:53.293 human work. Learning algebra by hand first is not about making you 00:13:53.293 --> 00:13:58.305 do busywork. It is about building the judgment that no software has. 00:13:59.533 --> 00:14:02.653 The fastest way to make any new idea stick is to use 00:14:02.653 --> 00:14:06.573 it within a few hours of first encountering it. Here are five 00:14:06.573 --> 00:14:11.433 things you can do today. First: look at any receipt. Every line 00:14:11.433 --> 00:14:16.073 item is an algebraic relationship — unit price times quantity equals the 00:14:16.073 --> 00:14:21.533 total for that item. Algebra is already running your grocery bill. Second: 00:14:21.533 --> 00:14:26.213 if you traveled somewhere today, try writing d equals s times t, 00:14:26.213 --> 00:14:31.733 where d is distance, s is speed, and t is time. Plug 00:14:31.733 --> 00:14:34.313 in two of the three and solve for the third — that 00:14:34.313 --> 00:14:39.413 is algebra in action. Third: find one situation today where you know 00:14:39.413 --> 00:14:43.593 a total and a rate but not a quantity, and write the 00:14:43.593 --> 00:14:46.513 equation for it. Do not solve it if you do not want 00:14:46.513 --> 00:14:51.953 to. Just write it. Fourth: open Desmos, which is free in any 00:14:51.953 --> 00:14:58.253 browser, and type y equals two x plus three. See the line. 00:14:58.253 --> 00:15:01.373 Now change the two to a five. See what happens to the 00:15:01.373 --> 00:15:09.153 line. Change the three to a negative one. Algebra becomes visible. Fifth: 00:15:09.153 --> 00:15:14.413 solve five x plus two equals twenty-seven. Write every step. Check the 00:15:14.413 --> 00:15:17.733 answer by plugging it back in. That is the full cycle of 00:15:17.733 --> 00:15:21.013 algebra, done once, with your own hand. 00:15:22.233 --> 00:15:29.893 Let us land the plane. Six things from today. One: algebra is 00:15:29.893 --> 00:15:35.953 reasoning about unknown quantities — a fundamentally different mental move from arithmetic, 00:15:35.953 --> 00:15:40.633 which computes with known quantities. Two: a variable is a placeholder for 00:15:40.633 --> 00:15:45.713 a specific number. The letter x does not mean 'anything' — it 00:15:45.713 --> 00:15:50.553 means 'the number that makes this equation true.' Three: an equation is 00:15:50.553 --> 00:15:54.373 a balance scale. The single rule of algebra is that whatever you 00:15:54.373 --> 00:15:58.213 do to one side, you must do to the other. Four: algebra 00:15:58.213 --> 00:16:04.093 was invented in ninth-century Baghdad by al-Khwarizmi, to solve practical inheritance and 00:16:04.093 --> 00:16:07.953 land disputes. It has been a professional tool for over a thousand 00:16:07.953 --> 00:16:15.613 years. Five: three properties — commutative, associative, and distributive — underpin every 00:16:15.613 --> 00:16:20.413 algebraic manipulation you will ever do. Six: always check your answer by 00:16:20.413 --> 00:16:24.693 substituting it back into the original equation. If both sides come out 00:16:24.693 --> 00:16:28.733 equal, you are done. If not, you made a mistake somewhere, and 00:16:28.733 --> 00:16:32.213 the check caught it before it mattered. Next class: we are going 00:16:32.213 --> 00:16:37.373 to dig into variables, expressions, and the art of simplification — how 00:16:37.373 --> 00:16:40.933 to take a messy algebraic expression and strip it down to its 00:16:40.933 --> 00:16:42.057 cleanest form.