Why is the distributive law correct in algebra, like in arithmetic?

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When they give reason for multiplication the negative numbers leading to positive number, they base on distribute law.But why the distribute law in algebra is correct like in arithmetic?(e.g why -5(8-6)=-5.8+-5.-6?).In abstract algebra they use distribute law as axiom.But in elementary algebra we must know reason for distribute law as a expanding of this in arithmetic.
 

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  • #2
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What is the difference between what you call algebra and arithmetic?

We started counting and ended up with the complex numbers. The distributive law occurred early with the integers. It is correct in the sense that it gives the results we are expecting.

The distributive law in abstract algebra comes as soon as we want to couple addition and multiplication, which is in a ring. Here we introduce it as axiom, since we are defining what a ring should be. The template were the integers, but a ring is the more abstract concept with many more examples. Hence the question about its validity doesn't come up. We defined it that way. How else should we justify something which didn't exist before?

As a matter of fact, the distributive law is a useful axiom which allows us to investigate all those different rings and algebras out there.
 
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  • #3
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The difference between arithmetic and algebra is: the distribute law in arithmetic is obvious when we use model of row and column(drawer model).But in algebra what do we interpret the -5 in -5(8-6)?
 
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Again, what is "in arithmetic" and what is "in algebra"? You only speak of integers, and the distributive law does what we expect, so we used it as axiom in other areas.
 
  • #5
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I mean negative and positive integers is algebra and positive integers is arithmetic
 
  • #6
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You need to read the historical development of numbers to understand where the notion of negative numbers come from and math has developed to what we have today.

Theres a book by Jan Gullberg that gets into it quite well.

Mathematics: From the Birth of Numbers

https://www.amazon.com/dp/039304002X/?tag=pfamazon01-20&tag=pfamazon01-20

The distributive law comes out of our desire to have multiple calculations have the same answer no matter which way is chosen.

So that 12x3+12x4 = 84 or I can sum 3 and 4 then multiply by 12 to get 12x(3+4) = 84
 
  • #7
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I mean negative and positive integers is algebra and positive integers is arithmetic
This is a wrong definition. Arithmetic is simply doing the numerical calculations to get an answer whereas algebra goes into the methodolgy of solving problems using variables, expressions, equations ... to solve a problem and get an answer.

Basically, if theres an unknown quantity to solve for its algebra. When you have all the numbers and can compute an answer its arithmetic.
 
  • #8
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The way is the following:
  1. Counting.
    This gives us the natural numbers.
  2. Zero.
    Someone in India some 5,000 years ago observed, that it is of great help to name something which isn't there: zero. My guess is it helped their accounting system for harvest, but we cannot know for sure.
  3. We now have what we call a semigroup. We can count and add. Of course the question quickly came up to balance accounting sheets. We therefore wrote positive numbers in one column, and what we nowadays call negative numbers in another column. Technically we extended our semigroup with additive inverse elements such that we could subtract, or as I assume: keep book of debts. We used the column system for really long. The Romans didn't have negative numbers. But at its core we had an additive group.
  4. Very early on calculations about the size of farm fields, projected sizes of harvest, taxes, etc. appeared in the accounting systems and we multiplied. The ancient Greeks also found geometry, which involves multiplications, too, but they weren't the first. Anyway, the distributive law for positive numbers is basically the addition of two rectangular fields which share a boundary.
  5. Now comes the trick: You could either say, that the distributive law for negative numbers must be as it is, in order to avoid contradictions in the system, or you can observe that a volume, and a length, are oriented quantities. It makes a difference whether you travel from LA to NYC or from NYC to LA. In one case you can visit the Empire State Building, in the other you cannot. Volumes are also oriented: clockwise and counterclockwise are different. I find the following picture best to explain the rules for negative numbers, which lead to their handling in the distributive law, too.

    1564024744576.png

    ##+ \cdot - = -\quad - \cdot - = +##
  6. It does what we expect it to do.
 
  • #9
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As a humorous aside, heres Abbott and Costello doing some alternative arithmetic


And heres a classic example of a teacher caught in the web of alternative math

 
  • #10
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So the rules of multiplication of negative or/and positive numbers lead to distribute law but not the distribute law lead to the rules of multiplication in your explanation?
 
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  • #11
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So the rules of multiplication of negative or/and positive numbers lead to distribute law but not the distribute law lead to the rules of multiplication in your explanation?
Yes and no. The distributive law is all which connects addition and multiplication.
Ok, let's do it formally. Say we want to solve ##(-5)\cdot (-7-8)##.
\begin{align*}
(1) \quad -5 \text{ is the solution of } &\quad x+5=0& 15\cdot x+75=0\\
(2) \quad -7 \text{ is the solution of } &\quad y+7=0 & 5y + 35 = 0\\
(3) \quad -8 \text{ is the solution of } &\quad z+8=0 & 5z+40=0
\end{align*}
With the help of the distributive law we get
\begin{align*}
0 &= (x+5)(y+7) \\
&= xy +7x+5y + 35\\
&= xy +7x \text{ by (2) } \\
&\text{ and equally }\\
0&= (x+5)(z+8)\\
&= xz +8x + 5z + 40\\
&= xz +8x \text{ by (3) }
\end{align*}
So ##\text{ by (1) }\quad 0=xy+xz + 7x + 8x = xy+xz + 15x = xy+xz + (-75)## which means ##xy+xz = 75##.
As ##(-5)\cdot (-7-8) = (-5)\cdot (-15) = x\cdot (y+z)=xy+xz##, we get with both ##(-5)\cdot (-15) = 75##.

So the distributive law can be used to show ##(-1) \cdot (-1) = +1## and similar the other rules.

If we vice versa have the rules for multiplying negative and positive numbers, we can show that the distributive law must hold:
##(-5)\cdot (-7-8)= 75 = 35+40 = (-5)\cdot (-7) + (-5)\cdot (-8)## or ##x\cdot (y+z) = xy +xz##.
 
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  • #12
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How can we explain the rule of multiplication of negative numbers for pupils of 6 grade?Should we use distributive law?And how should we explain for them the distributive law?
 
  • #13
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The difference between arithmetic and algebra is: the distribute law in arithmetic is obvious when we use model of row and column(drawer model).But in algebra what do we interpret the -5 in -5(8-6)?
The idea is perfect, and it's the same as I would have tried to explain. Symbols or plain numbers, Distributive Property works the same; no difference.
 
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  • #14
symbolipoint
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So the rules of multiplication of negative or/and positive numbers lead to distribute law but not the distribute law lead to the rules of multiplication in your explanation?
The jump from elementary school basic arithmetic to "high school" basic algebra makes critical use of the real number line to show signed numbers. Most of what was learned in elementary school becomes generalizable. The sign of the number makes little difference. Our rules become based on Addition and on Multiplication.
 
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  • #15
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As a humorous aside, heres Abbott and Costello doing some alternative arithmetic


And heres a classic example of a teacher caught in the web of alternative math

That second video stressed me out for some reason.
 

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