1.2 Rules for Addition (Basic Mathematics)
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- เผยแพร่เมื่อ 4 ส.ค. 2024
- This section covers how to add integers, positive or negative. Rather than tell you how it works,, Serge Lang uses proofs and identities to help you see how it works for yourself. Enjoy!
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I was really in a need for such playlists based on basics of Arithmetic. I really appreciate your hard work and your way of teaching.
Would you also be covering other maths books by Serge Lang such as his first course in calculus and linear algebra ? thanks and I’m very appreciative for your tutorials because I’m able to efficiently study this book now
Thank you for the resolution on the exercises, this helped a lot! Cheers!
Just received the book. As a high school student curious in pursuing further studies in physics, this is very helpful!
I am a primary school student
thank you for hard work and i really appreciate this
Good information teacher, thank.
Thank you so much...I really appreciate your time and your work.
Ikr
Awesome way to publish new videos!! Keep it up
I've got a backlog already! 2 videos right now, hopefully 1 more by the end of the day.
Thank you for making these videos. I'm a bit confused reading the book :))
you can solve 25:07 "-2-y=5" with this approach
-2-y=5
-2+2-y=5+2
simplifying
-y = 7
simplifying further: (i think i used the symmetric property)
-7 = y
commutative property
y = -7
/
-2-y+y=5+y
-2=5+y
-5-2=5-5+y
-5-2 = y
commutativity
y = -5-2
simplifying further:
y = -7
Thank you 👍
Surprisingly understandable
Brilliant!
For the proof at 19:00 I tried to come up with one myself before looking at the one in the book.
Using our axioms, if A and B are both positive integers their sum results in some other positive integer, C. To get the additive inverse of C, which will be guaranteed negative, we can subtract it from 0. So: - C = 0 - C.
knowing that C is the Sum of A and B:
- C = 0 - (A + B) = 0 + - ( A + B)
And as we proved just before this - (A+B) = -A - B so:
- C = 0 + (- A ) - B which is the same as -C = -A - B or -A + -B, proving that adding any two negative integers results in a negative.
Is this a valid way to go about proving this?
Thanks Sir
Do you mind telling me why are we supposed to rewrite numbers? I have always seen this but I never got it.
Thank you so much for these videos
Hi Jonathan. I have a question about your proof for N5. You state that , and after that you state that this comes from N3. However N3 is the converse of that statement, since it says . So, at 15:15, shouldn't you be saying "That's from equation N2" ?
Because I think that N2 and N3 are the converse of each other.
I think you're right. Good catch.
:) Thank you for your videos!
@@jg394 I just read your wiki and you consider that N3 is itself a "if and only if" kind of statement, do you think that is the correct way of interpreting it, or the way I said is how it should be interpreted? Because later in the book, in the Logic interlude, Lang says that sometimes he writes "if and only if statements" as regular "if ... then...". So now I don't know which way really is the correct way of thinking about this.
magathrea tuga when the “of and only if” statement comes into the book i was thrown off also(?), the rest of the first section I had no problems with but there was one specific equation that referred back to another one with an “it and only if statement”, i will have to go back and have a look again to see what through me off...
I’ll be back to report on it!
Sir, can I follow this video for my jee preparation? Plz tell sir!
Does the rule of adding two positive numbers result a positive breaks when we deal with things like Rammanujan series?
Great question. Rammanujan series, as you know, are divergent, meaning, they have no sum. (We have to do some calculus to calculate what the sum of an infinite series would be, and divergent series have no sum.) As a physics student, I find the idea entirely silly and something only a mathematician would find even remotely interesting. I think it's safe to say that when you're dealing with Rammanujan series, a whole lot of other things break, including the idea that adding two positive numbers should yield a positive number. It might also help to think of the -1/12(R) that is the result of the sum of all positive integers isn't a rational. It's a special number that doesn't belong in the rational set, and so it isn't even really negative, the same way integers and rationals and reals are negatives.
Thanks so much for this legendary service.
I have one real annoyance that is very basic. Maybe I'm going to deep into this, I understand basic algebra and am learning more about the proof side of things.
You even mention it, why can we add (-a) to both sides of the equation? Adding something to both sides isn't justified formally, it is kind of handwavey compared to how rigorous everything else is. 'Because it's an equation'
In simplest terms, the idea you are looking for is if "a = b" then "a + c = b + c". This should be obvious, because it would be truly strange if, say, 3 + 2 = 5" and "3 + 2 = 6" at the same time. The reason why it is "handwavey" is because it is inherent in the definition of what "=" means and what "+" means.
@@jg394 I sensed it was my own silliness in trying to adapt to the proof way of thinking. The if "a = b" then "a+c = b + c" is what I'm looking for. It's a bit tough to change the way you think towards more proof driven exercises, feels like a different game and at this early stage finding the right boundaries and scope is the hardest part, leads to dumb stuff like this :P
Anyway thanks for the videos and reply!
Hello can someone please help mw with question 4 of the exercise its goes like this (a - b) + (c - d) = (a + c) - (b + d) i dont understand where + comes from on the next to the d, wasn't it a negative before we rearrange the question. the answer i got was (a + c) - (b - d)
Great work 😮
Im so lucky to find you
Aside from math I would like to know where you put the camera
Serge writes his proof for N3 differently though. I don't understand the style in which he proves it. Like how does he come up with those expressions with two equals signs in them?
thank god
I did not get it at 25:17
if we get the result as y= -7, how can you add +7 to this equation (-2-y=5)
To get your equation, shouldn't we get y=+7?
Thanks...
I made a mistake there....
Edit: nope. The y is minus!
15:40 I have question regarding this part, why is that when you are to prove that - (a + b) = -a - b. why is there no minus sign outside of (a + b) in (a + b) + ( -a - b) = 0?
- (a+b) just moved to the other side of equal sign
That's why it became + sign
if - (a + b) = - a - b that means that - a - b is an additive inverse of (a + b) and the proof is (a + b) + (- a - b) = 0
At 23:24, if we don't flip the A but flip the 4 instead, can it be -1?
if you do so you will have :
-a = 3-4
-a = -1
a = 1
So this last rule where a + b can be a negative... can you explain? How does one just declare -n and -m as positive if there are minus signs there?
If m is a negative number, then -m is positive.
@@jg394 because -(-m) = m. Yes.
At 21:00, if you assign certain numbers to a, b & c, you get a contradictory answer on either side of the equation (a+b-b=c-b)
Weird.
It gets complicated with time or im getting dumb 🧠
Bro seriously???
Yes
Sir, i have a question at this part 19:33
You are saying that a = -n so why are you calling it positive n even though it has minus sign before it. Thank you very much.
"n" is taken to be positive by itself, that means it is a natural number.
But "a" is taken as the additive inverse of "n", that is, "-n".
For example, let n be 3, then a = -3. In that sense, n IS positive.
@@nibirsankar Thnks for answering the question. Btw may I ask one more question to you. Without any algebraic manipulation, how have we derived a=-(-a) from the equation N2 a+(-a)=-a+a=0.
I couldn't understand any math beyond simple things but your videos help me stay interested and understand math better, thank you so much 🤩🤩❤❤✨✨
I didn't understand the N3 :/ . Can someone help me? Why b=-a and a=-b?
a or b could be negative. For instance, if b is -3 and a is 3, then -a = b and -b = a
Sir for this topic I get confused when letters come in
What can be the solution
that can be any number. letter = (any number)
At 24:40 I just did: -2 - 5 = y then -3 = y. Am I wrong?
-2 -y = 5
-2 -5 = y
-(2+5)= y
-(7)=y
- 7=y
y=-7
it is better to go like this :
-2 -y=5
-2+2-y=5+2
0-y=7
-y=7
y=-7
I dont understand the n4
Some of these confuses me even more. In N4, you (or Serge) could've just said "This is true because - - = + and + + = +" Additionally, "- + = - and + - = -".
There is no thing like - - or + + on mathematics. Those signs aren't supposed to be used without other symbols.
Although what you're saying is common sense once you abstract their meaning, you can't write something incorrect in a book just for you to "teaching using a simple language".
"a = - ( - a)" can be used for substitution for you to solve the equations, while "- - = +" can't and doesn't makes any sense mathematically. I can see where you're coming from but unfortunately doesn't work that way and once you progress further on more complex topics you'll hopefully understand why.
I’m not gonna lie, I’m new to the world of physics and physics math, can someone explain the whole “If we add -b to both sides, will recover that a=-b”?
How? Are we just flipping the letters opposite or am I missing something…..feeling kinda dumb here..
On the left side, a+b becomes a+b-b or just a
I wasn't paying attention and when you got to the conclusion around 10:00 I thought the open parenthesis before -b was a C for far too long and had to keep rewatching earlier parts to see what I missed. I feel even dumber now, damn
👏
Thank you for this information. I'm an adult struggling with Math.
To solve last exercise (homework) i did: from a + b = a + c then b = c + a - a then b = c
guys im having some troubles with the proofs
Como assim?
same here, some are a bit confusing
So when we going over that homework
😘😘
How to solve this?
- (a+b+c) = -a+ (-b) +(-c)
thats -a-b-c the plus are pointless at that points because they're all negative (they're not minus unless you say that -a is positive)
why can't you just say that's because adding -a to a gives you 0, b has to be the same number as -a because it also gives you 0 when you add it to a? the proof makes it seem more complicated than simple logic
This is a great question. If you pay really close attention, you'll see how we use simple proofs using assumptions or conclusions we have already made to arrive at the fact that the additive inverse is unique. We'll use a similar strategy to prove that the multiplicative inverse is unique, and we'll do it for more than just numbers. Algebra isn't the study of numbers, it's the study of the OPERATIONS we can do with numbers. More advanced algebra doesn't even deal with numbers anymore -- things that behave more like combinations of a Rubik's Cube, or a point in space, or many other things, than a number.
That said, if you think you have a simpler proof, PLEASE SHARE IT. The proof you shared above doesn't give us the conclusion that -a = b. We need something prove definitively that there is no other possible b that can give us the same effect as -a.
If all this goes way above your head, don't worry too much. It will make sense when we've revisited this topic for the 14th time and we're looking at points, shapes, and matrices.
@@jg394 thanks for the reply! it really made a lot more sense once i saw that -a+0=0+b
Paper is free, not your attention ⚡
7:18
5:53 N3
20:12
Commutativity & Associativity Ha
Your wrong