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In mathematics we have the Addition operation from which subtraction and multiplication can be resulted. so it's not wrong to say they both have the concept of addition within them. $$ a*b = a+a+...+a \ \ \ \ \ (b\ times)$$ and $$ a-b = a+ (-b) .$$ division has the concept of multiplication within it: $$ \frac{a}{b} = a* \frac{1}{b}$$ but I'm not so sure if it is so, because there is still division in it in $ \frac{1}{b} $ and can't be decomposed any more as far as I know.

also, I assume division doesn't have the concept of addition in it: no way to say how $ \frac{a}{b} $ can be decomposed to addition, and also no way to say how it's possible to add "a", $ \frac{1}{b} $ times.

$ Q_1 $: so how is Division related to or can be derived from Addition and Multiplication?

$Q_2$ : what branch of math is this subject related to?

parvin
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  • In abstract algebra, there is a structure called a division ring that generalizes division from fields like $\mathbb{R}$ and $\mathbb{C}$ to a broader class of rings. – Michael L. Aug 01 '17 at 10:46
  • Yes; inverse operation. $a - a=0$ as well as $a \times \dfrac 1 a =1$. – Mauro ALLEGRANZA Aug 01 '17 at 10:46
  • Algebra? In a group $(G,\cdot)$, each element is invertible, i.e. there exists a neutral (identity) element $e\in G$ such that for every $g\in G$, there exists a unique $g'\in G$ such that $g\cdot g' = g'\cdot g = e$. Here, you want real numbers $(\mathbb{R}\setminus {0},\times)$ is a group, and the identity element is $1$.. – Clement C. Aug 01 '17 at 10:46
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    And the subject is arithmetic. – Mauro ALLEGRANZA Aug 01 '17 at 10:48
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    1/b is the inverse of b, it is usually stated as an axiom in say analysis, where every element of a field, say, has a multiplicative inverse. I would also say fields is a part of abstract algebra, which goes much further in depth than analysis. – smokeypeat Aug 01 '17 at 10:50
  • On Q1 concerning the link with addition: $\frac{a}{n}+\cdot+\frac{a}{n}=a$ (sum of $n$ terms on LHS). – drhab Aug 01 '17 at 10:51
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    You say that $\frac{a}{b}=a*\frac{1}{b}$ still has division in it, the exact same applies to $a-b = a + (-b)$, It's just that in this second notation we don't write the neutral element $0$ anymore, but you can: $a-b = a + (0-b)$. Both cases are examples of the exact same thing (note that in the first case, the $1$ is also the neutral element). yet in the case of substraction you are sure but in the case of division you are not? – Jens Renders Aug 01 '17 at 10:56

1 Answers1

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Division, as an operation, is the inverse of multiplication. So, the number $$\frac ab$$ is defined to be the unique solution to the equation $$b\cdot x = a.$$

This definition also explains why

  • $\frac a0$ is not defined for $a\neq 0$ (because there is no solution to the equation $0x=a$ in that case)
  • $\frac 00$ is not defined (because there is more than one solution to the equation $0\cdot x = 0$.
5xum
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  • thank you, so there is no division it's basically multiplication for numbers greater than 0 and less than 1. is that it? – parvin Aug 01 '17 at 10:54
  • if that's true, how can you explain 0.5 * b? add b half times? – parvin Aug 01 '17 at 10:55
  • @parvin Well sure there is such a thing as division. It's just that yes, division is the same thing as multiplication by the inverse. And the same thing as solving an equation. – 5xum Aug 01 '17 at 10:55
  • @parvin $0.5 b$ is the number that solves the equation $2x=b$. – 5xum Aug 01 '17 at 10:55
  • so it's just a mathematical concept, as you said. I was trying to explain it with the real world examples. – parvin Aug 01 '17 at 10:56
  • That first analogy was not quite right. If you think about $\frac{14}{7}$, you'll come unstuck. Think of division as doing multiplication backwards - we call that the inverse. If you want to change the term "divide by $x$" to "multiply by the inverse of $x$", that's your prerogative :-). – Matt Aug 01 '17 at 10:58
  • @MattS Which not-quite-right analogy are you talking about? – Arthur Aug 01 '17 at 10:59
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    @parvin You can have a real world example. I you have $b$ litres of milk, then $0.5\cdot b$ litres of milk is the amount of milk you must give each person if you want to split all the milk equally among $2$ people. – 5xum Aug 01 '17 at 11:00
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    @Arthur I was talking about the "there is no division it's basically multiplication for numbers greater than 0 and less then 1." analogy. – Matt Aug 01 '17 at 12:07