7

All positive and negative numbers including zero are called integers. So in the form $a=bq$, since $0 = 0ㆍq$ is true for any integer $q$, $0$ can have $0$ as a divisor of itself as well as a multiple of itself by the definition expressed by $a=bq$.

But why it's said "We cannot divide by $0$"? It's understood as "$0$ cannot be a divisor" to me.

"Definition: An integer a is called a multiple of an integer $b$ if $a=bq$ for some integer $q$. In this case we also say that b is a divisor of $a$, and we use the notation $b | a$ . . . On the other hand, for any integer $a$, we have $0 = aㆍ0$ and thus $0$ is a multiple of any integer."

Source: Abstract Algebra: Third Edition, John A. Beachy, William D. Blair, p.4.

"Rule Division by $0$ is undefined. Any expression with a divisor of $0$ is undefined. We cannot divide by $0$"

Source: Prealgebra: A Text/Workbook, Charles McKeague, p.61.

"Observe that division by the integer $0$ is not defined, since for $n≠0$ there is no integer $x$ such that $0ㆍx=n$ and since for $n =0$ every integer $x$ satisfies $0ㆍx=0$"

Source: Introduction to Mathematical Proofs, Second Edition, Charles Roberts, p.99.

[Now I understand my question more after reading number theory chapter of a book]

$0=d\cdot 0 $
Thus, 0 is a multiple of every integer except 0.

enter image description here

buzzee
  • 1,530
  • Is there a single unique value that zero divided by zero would be? – JB King Dec 28 '15 at 15:49
  • @JB King Should there be a single value of zero divided by zero so that zero can be a divisor? – buzzee Dec 28 '15 at 17:10
  • @buzzee, for all other integer divisions the answer $q$ is a unique value. To have a multi-valued solution does present some challenges compared to other operations. – JB King Dec 28 '15 at 17:31
  • @JB King So zero can't be a divisor because an infinite number of integer q satisfying 0ㆍq=0 is against that q is a unique value. But why 0 is a multiple of any integer if 0ㆍq=0 is not allowed? – buzzee Dec 28 '15 at 18:33
  • @Dietrich Burde I checked the link, but it doesn't solve my question. – buzzee Dec 29 '15 at 06:34
  • 4
    You are mixing terminology from three different books with very different focuses. One of them is pre-algebra - intended for students who may be 12 years old. Another is abstract algebra for college students. The underlying phenomenon is always the same, but the terminology that the books use to describe it varies depending on their audience and on the author's taste. – Carl Mummert Dec 29 '15 at 14:51
  • 1
    What do you mean by "$0\cdot q=0$ is not allowed"? We certainly are allowed to write $0\cdot q=0$ for any integer $q$, and this is how we know that $0$ is a multiple of $q$ for any integer $q$. It is also how we know that $0$ is a multiple of $0$ and that $0$ is a divisor of $0$. – David K Jan 04 '16 at 05:47
  • The comment of @CarlMummert has said it all, but let me add one remark: when we say “$m$ divides $n$” or “$n$ is divisible by $m$”, we are most certainly not talking about the operation of division (÷). – Lubin Mar 06 '17 at 22:34

1 Answers1

8

Every integer divides zero, including zero itself; however, the only integer that zero divides is itself. That is, $b \mid 0$ for all integers $b$; but if $a$ is an integer and $0 \mid a$, then $a = 0$.

When it is said that "you can't divide by zero", what is meant is that, given an integer $a$, there is not a unique quotient upon division by zero.

Specifically, given integers $a$ and $b$, with $b \ne 0$, if $b \mid a$ then there is a unique integer $q$ such that $a = qb$, namely $q=\frac{a}{b}$. However, if we allow the case when $b=0$, then we lose the uniqueness. Indeed, as already mentioned, $0 = q \cdot 0$ for all integers $q$, so it makes no sense to assign a value to the expression $\frac{0}{0}$. And if $a \ne 0$ then there is no integer $q$ such that $a = q \cdot 0$, so it also doesn't make any sense to assign a value to the expression $\frac{a}{0}$.

Clive Newstead
  • 64,925
  • So is the statement "We cannot divide by 0" wrong? – buzzee Dec 28 '15 at 17:07
  • 2
    @buzzee It's not wrong. It's just that "$a$ is divisor of $b$" and "$b$ can be divided by $a$" mean two different things. – Wojowu Dec 28 '15 at 17:39
  • @Wojowu Can I get a more detailed explanation of how two statements are different? I thought they all can be expressed by "a=bq", where a is a multiple of b and, b is a divisor of a when we let a, b, q are integers. – buzzee Dec 28 '15 at 17:56
  • 1
    @buzzee We define $a/b$ to be the unique number $q$ such that $a=bq$, if it exists. For $b=0$, either existence or uniqueness fails. – Wojowu Dec 28 '15 at 18:03
  • @Wojowu You mean by "a/b", 'a can be divided by a', right? But, isn't b a divisor of a in both "a/b" and "a=bq"? – buzzee Dec 28 '15 at 18:17
  • 1
    @buzzee It is. The problem is with other direction. If we know that $a$ is divisor of $b$, then we can't conclude $a/b$ exists. For example, take $a=b=0$. $0$ is a divisor of itself, but there is no unique number $q$ giving $0=0q$. Hence $0/0$ doesn't exist. – Wojowu Dec 28 '15 at 18:20
  • A)" b can be divided by x" means b/x $\in \mathbb Z$. B)"x is a divisor of b means "q$\cdot$ x = b" for some q $\in \mathbb Z$. These are two different statements and mean different things. A $\implies$ B but B $\not \implies$ A. If x $\ne$ 0 then B $\implies$ A but if x = 0 then B is true but A is not. – fleablood Jan 04 '16 at 06:12
  • "We can not divide by 0" is true. "0 is not a divisor of any number except 0" is true. "0 is a divisor of 0" is true. "If we can divide by x, then x is a divisor" is true. "If x is a divisor, then we can divide by x" is !!!!!!!!NOT!!!!!! true. "If x is a divisor, then we can divide by x if x $\ne$ 0" is true. – fleablood Jan 04 '16 at 06:15