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Why all the letters in algebra?

In algebra, we often want to work with general concepts instead of specific numbers. Using letters allows us to represent quantities that could change or that we don't know the value of yet. Letters can also be used to generalize formulas or equations. Created by Sal Khan.

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  • male robot hal style avatar for user Giles
    At , Sal mentions the convention of using letters comes from history. Where in history did the convention of letters come from, and how did x and y become so popular (especially with the coordinate plane?)
    (415 votes)
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    • leafers sapling style avatar for user Peter Collingridge
      There's a short TED talk about why x became the standard letter representing an unknown: http://www.youtube.com/watch?v=YX_OxBfsvbk. In short it is the English (or Latin) version of the Greek letter χ (chi), which was the first letter of the transliteration of the Arabic word for 'something', which was used in the original algebra texts.

      Since x was commonly used for one unknown, it made sense to use y as a second, and z as a third unknown.
      (530 votes)
  • ohnoes default style avatar for user Microwaved Chicken Wing
    At , Can we use any symbol (ë, æ, ´¬ or ¤) represent an unknown number?
    (11 votes)
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  • duskpin sapling style avatar for user Ishy Ish
    why do we need to have letters
    why can't they just give it to us?

    EDIT: Well, I actually meant to ask what's the use of a variable and how is it used in real life.
    (6 votes)
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    • piceratops seedling style avatar for user ViseanTourist
      Because the letters are for when you don't know something - in real life, say you wanted to know how much it cost to drive your car somewhere, and you knew that gas was $5 a gallon, and that your car got 60 miles to the gallon. With that knowledge you can write an equation with x equalling how many miles you drive, and you can use that equation again and again to figure out how much it would cost you to go different distances.

      If you didn't have the x in there, you would have to create the equation each time with a different distance.
      (12 votes)
  • blobby green style avatar for user mcgaritylibbyclaire
    Why do we have hard hard math
    (8 votes)
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    • sneak peak blue style avatar for user Leo Pi
      1. Just because it's hard for you doesn't mean it's hard for everyone.
      2. Maths the language of the universe, and the laws of the universe don't have to be simple to understand for them to be true.
      (12 votes)
  • aqualine ultimate style avatar for user BungoNisekoi
    Imagine you are a person who doesn't know any languages that use the Latin alphabet letters (ex. a, b, c). Do you do algebra with letters from the Latin alphabet, or do you do algebra with letters from your own non-Latin based language?
    (7 votes)
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  • mr pink red style avatar for user 41sweetworld
    Why was Jesse Roe in this video if Sal was going to do all the talking anyway? :)
    (10 votes)
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  • cacteye purple style avatar for user Ibrahim Sabbir
    This would be really helpful for some students. While I was a student my teachers didn't really want to answer that question. why? because he thought this was easy. But I was first doing this stuff. I needed answers.
    (9 votes)
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  • leaf grey style avatar for user 栄耀♪
    While Sal was explaining the second reason to why we use letters(), he wrote and used the equation y = x + 1 . But did he actually mean to use y = y + 1 ? The examples he gave were 3 -> 4 , 5 -> 6 , 7.1 -> 8.1 . So the pattern here is whatever you give, I'll give that amount +1 . So wouldn't the correct equation be y = y + 1 ? Or am I just confusing this with coding(variable declaration). (I am an active coder)
    (6 votes)
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  • marcimus orange style avatar for user abigail.rogers
    how does it work it seams complicated how would you describe it
    (3 votes)
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    • blobby green style avatar for user Stephen Waltz
      I had the same reaction and would have preferred simpler examples.
      Sal actually does do something like I do briefly in the next video.
      You are probably familiar with problems like this.
      4 + 3 = _
      In Algebra we might write it like this and ask the value of a instead of what number goes in the blank.
      a = 4 + 3.
      If you think that just makes it harder, I can't disagree.
      Even this might seem simple.
      7 = 4 + _

      While this seems harder.
      7 = 4 + a
      This seems easy.
      3 x 5 = _
      This seems harder.
      a = 3 x 5
      Even this may seem easy.
      15 = 3 x _

      This may seem harder (especially without the multiplication symbol).
      3a = 15
      What about something like this?
      3 x __ - 4 = 5
      This kind of problem is where variables might start to make sense.
      3a - 4 = 5
      You may still be able to solve the problem easily because the numbers are small and the answer is an integer.
      Consider this problem.
      Tom needs $47 more to buy a pair of shoes he wants,
      He has a chance to earn some money painting a fence.
      He figures the job will take 4 hours and the paint will cost $20.
      If he is right, what is the least hourly rate he can charge to earn the money he needs.
      You might be able to figure this out without using Algebra (and a variable) but it won't be easier.
      (6 votes)
  • blobby green style avatar for user ruther0217
    las letras en las matematicas son basicamente variables independientes que sirven para cambiar el resultado de una ecuacion ?
    (4 votes)
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Video transcript

I'm here with Jesse Ro, whose a math teacher at Summit San Jose and a Khan Academy teaching fellow and you had some interesting ideas or questions. Yeah, one question that students ask a lot when they start Algebra is why do we need letters, why can't we just use numbers for everything? Why letters? So why do we have all these Xs and Ys and Zs and ABCs when we start dealing with Algebra? Yeah, exactly. That's interesting, well why don't we let people think about that for a second. So Sal, how would you answer this question? Why do we need letters in Algebra? So why letters. So there are a couple of ways I'd think about it. One is if you have an unknown. So if I were to write X plus three is equal to ten the reason why we're doing this is that we don't know what X is It's literally an unknown. And so we're going to solve for it in some way. But it did not have to be the letter X. We could have literally written blank plus three is equal to ten. Or we could have written Question Mark plus three is equal to ten. So it didn't have to be letters, but we needed some type of symbol. It literally could've been Smiley Face plus three is equal to ten. But until you know it, you need some type of a symbol to represent whatever that number is. Now we can go and solve this equation and then know what that symbol represents. But if we knew it ahead of time, it wouldn't be an unknown. It wouldn't be something that we didn't know. So that's one reason why I would use letters and where just numbers by itself wouldn't be helpful. The other is when you're describing relationships between numbers. So I could do something like - I could say - that whenever you give me a three, I'm going to give you a four. And I could say, if you give me a five, I'm going to give you a six. And i could keep going on and on forever. If you give me a 7.1, I'm going to give you an 8.1. And I could keep listing this on and on forever. Maybe you could give me any number, and I could tell you what I'm going to give you. But I would obviously run out of space and time if I were to list all of them. And we could do that much more elegantly if we used letters to describe the relationship. Maybe what you give me we call X, and what I give you we call Y. And so I say, look, whatever you give me, I'm going to add one to it. And that's what I'm going to give back to you. And so now, this very simple equation here can describe an infinite number of relationships between X or an infinite number of corresponding Ys and Xs. So now someone knows whatever X you give me you give me three, I add one to it, and I'm going to give you four. You give me 7.1, I'm going to add one to it and give you 8.1. So there is no more elegant way that you could've done it than by using symbols. With that said, I didn't have to use Xs and Ys. This is just a convention that kind of comes to use from history. I could've defined what you give me as Star and what I give you as Smiley Face and this also would've been a valid way to express this. So the letters are really just symbols. Nothing more.