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### Course: Integrated math 1 > Unit 1

Lesson 1: Overview and history of algebra# Abstract-ness

Abstraction is like taking a real thing and turning it into an idea. We can use a cube as an example. Even though cubes can look different in real life, the abstract idea of what a cube is stays the same. Created by Sal Khan.

## Want to join the conversation?

- At3:17Sal said that in reality there is no perfect cube. Is there any perfect 3-dimensional objects in reality (ex. perfect spheres, perfect cubes, perfect rectangular prisms, perfect pyramids, etc.)?(39 votes)
- In the real world, we also have something like "spacetime", which is sometimes seen as the fourth dimension. In mathematical abstracts, even solid figures are represented without any specific reference to spacetime. Due to that, there cannot exist any ideal "cube" for example. Any real cube will include some aspects and features that are specific to them. That's a philosophical problem as well.(35 votes)

- Wait, why is Sal talking about Geometry at2:37? I thought this was an Algebra class...

,:\(18 votes)- While algebra has to do with equations and formulas, geometry has to do with objects and shape. It may not seem like they're related, but they actually are.

Take y = x as an example. This equation can be graphed so that it creates a set of point that turn into a straight line. An equation, an algebraic concept, can be graphed so that it becomes a geometric concept!

Another example would be the Pythagorean Theorem (a^2 + b^2 = c^2), a theorem that states if the legs of length "a" and "b" are of a right triangle then it's hypotenuse would be of length "c". Here, the side lengths of a right triangle (a geometric concept) can be then related to an equation (a algebraic concept).(46 votes)

- How and why is algebra so abstract?(18 votes)
- Something that is abstract is something that exists in your thoughts and your mind but is not a physical object. That said, algebra is abstract because numbers are a man-made metaphysical construct used to do mathematics.(31 votes)

- to wrap up:

Abstract is Abstract(25 votes)- Thnx 4 making it so Simple LOL(4 votes)

- i dont understand the comparison between abstractness and real world and reality, when the definition of abstract is theoretical ideal or theory. My question is why would we base our devoted time and brain capacity for something we still have yet to prove its signfigance to our everyday life and prove further from theory?(11 votes)
- Abstraction is studied as it's the process of compressing information in our brain to make room for more concepts. Imagine how much devoted time and brain capacity would have to be wasted if we were to remember every single pointless minute detail of every math problem.

Instead, we develop theories, equations, formulas that can be applied to multiple instances of math. Imagine how difficult it would be to remember the quantity of every single object if we were not to use a numerical symbol to represent it such as 1, 2 or 3. Imagine how much time we'd waste pointing at various objects to try to convey an idea rather than saying abstract words that represent them instead such as "wood", "stone", "food", etc.(22 votes)

- what is the purpose of making about video Abstract-ness? Is really helpful in learning algebra?(9 votes)
- Up until now, you have been working with specific numbers as you learn basic arithmetic. In algebra, you will be using variables to generalize calculations or to represent unknown values. You won't know what number the variables represent. This is where the abstraction come in. You will learn how to simplify algebraic expressions to the extent possible even though you don't know the value of the variables.(25 votes)

- i am curious that what mathematical rules or concepts are taken from early mathematicians ?(10 votes)
- Apoorva, all of math is derived from the foundational basics. It is absolutely essential that you understand the foundational concepts to build on the higher mathematical rules a head.(11 votes)

- Would this be considered a good definition of "abstract:"

"Abstract: considered apart from concrete existence."?(13 votes)- i will report you to teacher(4 votes)

- why are keep saying abstract(11 votes)
- Is this Algebra 1 course for grade 9 students?(10 votes)
- yeah. that's why it's under mathematics 1(2 votes)

## Video transcript

You will hear me use the word abstract a lot so i thought i would actually give you an attempt at a definition, or maybe even more important an intuition of what abstract means and abstract can be an adjective you can have an abstract idea you can have abstract art or it can be a verb you can abstract something abstract the idea from some other idea and you can even have it as a noun you can have an abstract and it tends to, if you use it as a noun the one I tend to associate is the abstract of research paper, which is kind of, distills the essence of the research paper, which is kind of a summary of that paper and the one thing you are going to see regardless of how you what context you use the word abstract there's this kind of notion taking the essence of a real world object whether you use it as a noun, adjective, or a verb so over here we have our real world we have our real world and then over here you have your world of you have ideas and concepts and the general idea behind the abstraction or abstracting something is that you're taking it away from the particular concrete real world and you're going more into the direction of ideas and concepts and probably for me, one of the most tangible ways of thinking about abstraction, which is kind of a contradiction in itself to think of abstraction in a tangible way is things like geometric shapes so if i were to tell you to find me some cubes you might point to a borg vessel right over there a borg vessel you might point to a pair of dice let me draw a pair of dice if you were looking for cubes so you might point to a pair of dice that looks something like that you might point to a Rubik's cube anything you might find there might be a building that looks like a cube it might be a building that looks like a cube or maybe there is a box in your house that is a cube but in your mind you have a general idea of what a cube is like i know a cube when i see one and a general idea is distilling the concept the idea of what a cube is and all of these ideas are very different this is some plastic thing i could hold in my hand these are these white things they aren't even geometrically close to being perfect they have these little divets on the side right over there this is a large borg vessel that you know that doesn't exist yet is a fictional thing that they all have this cubeness to them one of the fun things about geometry so really distill the essence of these real world shapes and we do have this definition in geometry which is an object like this where every side has the exact same length so this is length one and that would be length one that would be length one it doesn't have to be whatever the length of this side is and this dimension then this dimension would be that length and then that dimension is going to be that length and i'm not giving you the rigorous definition but i'm just trying to highlight that there is pure idea of what a cube is of what a cube is and the real world, there is nothing that is actually a perfect cube if you were to get really really close to the die if you were to measure exactly their measurements they won't be exactly the same measurement but the abstract idea is completely the same length as this and this and this and all of the edges are going to have the exact same length so this is going from the concrete the specific from the real world if you consider the 24th or the 25th century, the real world to going to the idea behind it, the general idea and you probably also heard the word abstract in terms of art, like abstract art so this is abstract art and it is the same general idea so if you to look it up in the dictionary you're going to find 20 definitions of the word abstract but it's all, essentially trying to say the same thing abstract art is art that is not focused on trying to paint reality the exact way reality exists if you look at kind of lot of Renaissance art they are skilled at painting figures exactly how they look in the real world but the abstract artist sometimes they're not even trying to represent anything from the real world they are trying to represent a raw idea or a raw expression of color and form and texture and this is a Jackson Pollock painting right over here and i printed out so many things wrong taken by our own Steven Zucker, our art historian and you can see that it's not clear Jackson Pollock is not trying to paint a you know, a dog or horse or anything like that he is painting something that is devoid completely independent of anything that we actually see in physical reality and the word, abstraction, you know, it doesn't just apply on just a pure geometry and art it applies to almost everything we do on a daily basis when we even talk about things when we even use words or use symbols we are essentially abstracting away we're abstracting the essence of something that actually exists in physical reality so if i use the word dog it is a set of symbols that represent something in our mind that we associate with dog we have in our minds kind of the quality of what a dog actually is you know, it has four legs and floppy ears and you enjoy petting it and they're man's or i guess people's best friend you imagine this thing called a dog and it has the essence of dog and when you actually look at dogs in the real world they look very very different type of animals like when you look at a great dane or kind of a super small poodle But we recognize there is an essence of those particulars that we can abstract away and say this is a dog and we abstract it even more by representing these letter symbols that tend to conjure up this image even when we write something as simple as a number so if i write the number five we use it so frequently that to us a number 5 seems kind of like a concrete thing but it 's so abstract it's just a quantity of things i can symbolize like that i could have symbolized 5 like that i could have symbolized 5 in roman numerals like that i could symbolize it like that and in all of these cases they are it is the idea of a quantity of five things five, you know, you could say point me to a five and someone could draw or point you to something like that but they are still pointing you to the symbol of five but it's still an very abstract idea so hopefully this gives you an appreciation for what abstract means as you can tell, you know it's kind of a, for lack of a better word it's kind of an abstract idea not to be too cute about that