Why we cannot divide by zero
Michael J. Neely
University of Southern California
Michael J. Neely
University of Southern California
http://www-bcf.usc.edu/˜mjneely
These notes discuss why we cannot divide by 0. The short answer is that 0 has no multiplicative inverse, and any attempt
to define a real number as the multiplicative inverse of 0 would result in the contradiction 0 = 1. Some people find these
points to be confusing. These notes may be useful for anyone with questions about dividing by 0. In order to ask why the
division operation 1/0 is not defined, it is important to understand where division comes from in the first place. Thus, the first
part of these notes discuss axioms of arithmetic. Readers looking for quick insight and intuition, without all the mathematical
overhead, can skip to the brief sections D, E, F.
A. Axioms
Arithmetic starts by assuming there are objects called real numbers. It assumes these real numbers can be manipulated by
operations of addition and multiplication that take two real numbers and produce a third:
• Addition: For any two real numbers a and b, there is a real number a + b.
• Multiplication: For any two real numbers a and b, there is a real number ab.
These addition and multiplication operations are assumed to satisfy certain axioms, called axioms for a field. The axioms
include standard commutative, associative, and distributive laws for arithmetic. The complete list of axioms is below:1
1) Commutative law: a + b = b + a and ab = ba.
2) Associative law: (a + b) + c = a + (b + c) and (ab)c = a(bc).
3) Distributive law: a(b + c) = ab + ac.
4) Existence of 0: There is a number “0,” called the additive identity, that satisfies a + 0 = a for all real numbers a.
5) Existence of 1: There is a number “1,” called the multiplicative identity, that satisfies a1 = a for all real numbers a.
6) Additive inverse: For every real number a, there is a real number －a, called the additive inverse of a, that satisfies
a + －a = 0.
7) Multiplicative inverse: For every nonzero real number a, there is a real number a
－1
, called the multiplicative inverse of
a, that satisfies aa－1 = 1.
8) 0 6= 1.
The axioms cannot be proven: They are a short list of properties that we intuitively expect numbers to satisfy. However, the
axioms are useful in that, from them, important additional facts about numbers can be proven. An important theorem that can
be proven directly from the axioms is that multiplying any real number by 0 produces 0 as the result.
Theorem 1: For every real number x, the following equation is true:
0x = 0
Proof: See Appendix A for the standard proof.
B. Criticizing the multiplicative inverse axiom
One immediately notices an asymmetry in the way the additive and multiplicative inverse axioms above are stated (axioms
6 and 7 in the above list). The additive inverse axiom holds for all real numbers, while the multiplicative inverse axiom holds
only for all nonzero real numbers. Why is 0 excluded? This is not fair! For a direct answer to this question, the reader can
skip to Section D. Readers who prefer the question “why can’t we divide by zero?” must admit that this question cannot even
be asked until we define division, which is done in the next section.
C. Subtraction and division defined as inverse operations
The axioms are stated in terms of addition and multiplication operations. Operations of subtraction and division are defined
after the axioms are stated. Here are the standard definitions.
• Subtraction: For two real numbers a, b, the subtraction a － b is defined as a + －b.
• Division: For two real numbers a, b, with b nonzero, the division a/b is defined as ab－1
.
In particular, 1/2 is defined as 2
－1
and 1/5 is defined as 5
－1
. Thus, 1/0 is not defined, simply because there is no 0
－1
.
That is, 0 is not defined to have a multiplicative inverse.
1The 8th axiom “0 6= 1” can equivalently be replaced by the assumption “there exists a nonzero real number.” Neither of these statements can be proven
without the other, since the case of all numbers being 0 is consistent with the 7 other axioms. Strictly speaking, the real numbers require more structural
assumptions to distinguish them from other fields that satisfy the 8 axioms. That additional structure is not relevant to these notes and hence is omitted.
2
As an example, if bd 6= 0 then the expression ad+bc
ad is defined as:
ad + bc
bd = (ad + bc)(bd)
－1
This is the multiplication of the “numerator” (ad + bc) and the multiplicative inverse of the “denominator” bd. The value
(bd)
－1
is well defined since bd is assumed to be nonzero. It can be proven from the axioms that bd 6= 0 if and only if b 6= 0
and d 6= 0. Thus, if bd 6= 0 then both b
－1
and d
－1
exist. This can be used to derive standard rules for manipulating fractions
such as (see Appendix B for the derivation):
a
b
+
c
d
=
ad + bc
bd
which holds if b 6= 0 and d 6= 0 (else, one or more of the expressions are not defined). Another standard rule is:
1
b/c =
c
b
which holds if b 6= 0 and c 6= 0 (else, one or more of the expressions are not defined).
D. The contradiction if 0
－1
is defined
From the definition of division, the following questions are equivalent:
• Why can’t we divide by 0?
• Why is the expression a/b undefined if b = 0?
• Why is the expression 1/b undefined if b = 0?
• Why doesn’t 0 have a multiplicative inverse?
Let’s focus on the last question. Why don’t we define a particular number to be the multiplicative inverse of 0? Let’s suppose
we choose a particular real number z that we define as the multiplicative inverse of 0, so that z = 0－1
. Thus, by definition of
a multiplicative inverse:
0z = 1
However, we also know from Theorem 1 that 0z = 0. This means 1 = 0, a contradiction. Thus, we cannot assign any real
number to be 0
－1 without reaching a contradiction in our pre-established facts.
E. Let’s make z a non-real number
We could define z = 0－1
to be a new symbol, not a real number. But in that case, we have to admit that this new symbol
z may not obey the rules of arithmetic that real numbers do.
Once we admit that z may not obey the rules of arithmetic, we have the complicated task of figuring out which rules it does
satisfy without leading to a contradiction. Even if we did that, we would have to figure out a reason why we are introducing
such a number z. What is the use of it?
Since there is no apparent usefulness of such a non-real number z, it is easier to simply avoid trying to define it. Then, we
just admit that 0 has no multiplicative inverse.
F. Intuition on reversibility
Here is an intuitive reason why 0 should not have a multiplicative inverse: An invertible operation should be “reversible” in
the following sense:
If I take a number x and multiply by 5, I get 5x. I can “reverse this” by multiplying by 5
－1
(equivalently, dividing by 5)
to get x back. Multiplying by 5 does not lose any information about the original number x. For example, if x is unknown,
but I tell you I multiply x by 5 and the result is 10, you can determine what x was just by solving the equation 5x = 10.
The invertible operations of multiplying and dividing by 5 are like a photocopy machine that shrinks or expands a document
by a factor of 5 depending on which button is pushed. We can shrink and expand as many times as we like, with no loss of
information.
Now let’s take an unknown number x and multiply by 0. The result is 0. I cannot recover x, all I have is 0. The equation
0x = 0 does not offer any information about x. The photocopy machine reduced my document to nothing! Multiplying by 0
is a noninvertible operation! We cannot go back by “dividing by 0.”
3
G. Intuitive limit arguments
It can be shown that if n is a nonzero number, then (1/n)
－1 = (n
－1
)
－1 = n. Now let n be a very large positive number.
We intuitively understand that 1/n approaches 0 as n is increased to ∞. Thus, we intuitively expect (1/n)
－1
to approach 0
－1
as n is increased to ∞ (assuming 0
－1
is defined in some form). But (1/n)
－1 = n, which has a limit of ∞ as n is increased
to ∞. This suggests that 0
－1 = ∞.
If this argument convinces you to define 0
－1
as the non-real number ∞, think again. By a similar argument it holds that
1/n approaches 0 as n is decreased to －∞ (such as when n is chosen as －10000 and then －1000000000). Thus, intuitively
one expects (1/n)
－1
to approach 0
－1
as n is decreased to －∞. But (1/n)
－1 = n has a limit of －∞ as n approaches －∞.
This suggests that 0
－1 = －∞. How can 0
－1 be both ∞ and －∞?
APPENDIX A—PROVING THEOREM 1
The axioms are stated with an implicit understanding of the notion of an equation such as a(b + c) = ab + ac. The equation
means that the real number a(b+c) is the same as the real number ab+ac. If an equation is “true,” then another true equation
can be obtained by adding the same thing to both sides, or by multiplying the same thing to both sides.2 This is used in the
following proof. The proof also uses notation x1 to represent the multiplication of a real number x and the number 1.
Proof: (Theorem 1) We know from axiom 4 that:
1 + 0 = 1
Multiplying both sides by x gives:
x(1 + 0) = x1
Using the distributive law gives:
x1 + x0 = x1
Using the fact that x1 = x gives:
x + x0 = x
Adding －x to both sides gives:
－x + (x + x0) = －x + x
Using the associative law:
(－x + x) + x0 = －x + x
Using the fact that －x + x = 0 gives:
0 + x0 = 0
Using the fact that 0 + x0 = x0 gives:
x0 = 0
Using the commutative law x0 = 0x proves that 0x = 0.
APPENDIX B—STANDARD RULES OF FRACTIONS
Recall that if bd 6= 0, it can be proven from the axioms that b 6= 0 and d 6= 0, and so both b
－1
and d
－1
exist. Further, if
bd 6= 0, it can be proven from the axioms that (bd)
－1 = b
－1d
－1
. Now suppose that bd 6= 0. Then b 6= 0 and d 6= 0, and so:
ad + bc
bd = (ad + bc)(bd)
－1
= (bd)
－1
(ad + bc)
= b
－1
d
－1
(ad + bc)
= b
－1
d
－1
ad + b
－1
d
－1
bc
= ab－1
(dd－1
) + cd－1
(bb－1
)
= ab－1 + cd－1
=
a
b
+
c
d
which is the derivation of the standard rule a
b +
c
d =
ad+bc
bd for manipulating fractions.
Other rules such as 1
(b/c) =
c
b
are obtained from the fact (a
－1
)
－1 = a (which again can be proven from the axioms),
provided that a, b, and c are nonzero.
The axioms can also be used to prove that for any real number a, we have －(－a) = a and －a = (－1)a. In particular,
1 = －(－1) = (－1)(－1).
2Manipulations of adding and/or multiplying an equation by the same thing can be justified from the more basic concept of substituting into an equation,
but that detail is tangential and is not discussed here.http://ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf
再生核研究所声明297(2016.05.19) 豊かなゼロ、空の世界、隠れた未知の世界
ゼロ除算の研究を進めているが、微分方程式のある項を落とした場合の解と落とす前の解を結び付ける具体的な方法として、ゼロ除算の解析の具体的な応用がある事が分かった。この事実は、広く世の現象として、面白い視点に気づかせたので、普遍的な現象として、生きた形で表現したい。
ある項を落とした微分方程式とは、逆に言えば、与えられた微分方程式はさらに 複雑な微分方程式において、沢山の項を落として考えられた簡略の微分方程式であると考えられる。どのくらいの項を落としたかと考えれば、限りない項が存在して、殆どがゼロとして消された微分方程式であると見なせる。この意味で、ゼロの世界は限りなく広がっていると考えられる。
消された見えない世界は ゼロの世界、空、ある隠された世界として、無限に存在していると考えられる。たまたま、現れた項が 表現する物理現象を記述していると言える。
これは、地球に繁茂する動植物が、大海や大地から、生まれては、それらに回帰する現象と同様と言える。大量に発生した卵の極一部がそれぞれの生物に成長して、やがて元の世界に戻り、豊かな大海や大地は生命の存在の元、隠れた存在の大いなる世界であると考えられる。無数の生命の発生と回帰した世界の様は 生物、生体の様の変化は捉えられても、人間の精神活用や生命の生命活動の様の精しい様などは 殆ど何も分からない存在であると言える。我々の認知した世界と発生して来た世界と消えて行った認知できない世界である。
このような視点で、人間にとって最も大事なことは 何だろうか。それは、個々の人間も、人類も 大きな存在の中の小さな存在であることを先ず自覚して、背後に存在する大いなる基礎、環境に畏敬の念を抱き、謙虚さを保つことではないだろうか。この視点では日本古来の神道の精神こそ、宗教の原点として大事では ないだろうか。未知なる自然に対する畏敬の念である。実際、日本でも、世界各地でも人工物を建設するとき、神事を行い、神の許しを求めてきたものである。その心は大いなる存在と人間の調和を志向する意味で人間存在の原理ではないだろうか。それはそもそも 原罪の概念そのものであると言える。
しかしながら、人類が好きなように生きたいも道理であり、巨大都市を建設して、環境を汚染して生存を享受したいも道理であるから、それらの一面も否定できず、それは結局全体的な有り様の中でのバランスの問題ではないだろうか。人類の進化の面には必然的に人類絶滅の要素が内在していると考えられる：
再生核研究所声明 144（2013.12.12） 人類滅亡の概念 － 進化とは 滅亡への過程である
そこで、結局は全体的な調和、バランスの問題である：
再生核研究所声明 56： アースデイ の理念
発想における最も大事なことに触れたが、表現したかった元を回想したい。― それは存在と非存在の間の微妙な有り様と非存在の認知できない限りない世界に想いを致す心情そのものであった。無数とも言える人間の想いはどこに消えて行ったのだろうか。先も分からず、由来も分からない。世の中は雲のような存在であると言える。
以 上
何故ゼロ除算が不可能であったか理由
１ 割り算を掛け算の逆と考えた事
２ 極限で考えようとした事
３ 教科書やあらゆる文献が、不可能であると書いてあるので、みんなそう思った。
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