Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Zero (constant)"
m (Added Categories)
 
(15 intermediate revisions by 5 users not shown)
Line 2: Line 2:
  
 
The development of a concept and notation for 0, probably in ancient Indian civilization, and its subsequent transmission to Europe via the Persians and Arabs, was fundamental to the success of western mathematics in fields beyond [[geometry]].  It has suprisingly much relevance due to its significance in [[positional number system]]s.  For instance, normal commercial interactions might be seriously slowed if cashiers had to make change on a purchase of LXIV dollars with bills marked L, X, V and I when handed XC dollars.
 
The development of a concept and notation for 0, probably in ancient Indian civilization, and its subsequent transmission to Europe via the Persians and Arabs, was fundamental to the success of western mathematics in fields beyond [[geometry]].  It has suprisingly much relevance due to its significance in [[positional number system]]s.  For instance, normal commercial interactions might be seriously slowed if cashiers had to make change on a purchase of LXIV dollars with bills marked L, X, V and I when handed XC dollars.
 +
 +
== Why Teachers warn you not to divide by 0 ==
 +
Teachers keep saying, " Do not divide by zero." You might be thinking, Why? First, normally dividing by a smaller number gives you bigger results. So maybe <math>1/0</math> equals ∞! But that is not the case. 1 over <math>x</math> is the reciprocal or the multiplicative inverse of
 +
<math>x</math>. So anything times its reciprocal has to be 1. Therefore, 0 has no reciprocal because anything times 0 is 0.
  
 
== Operations with 0 ==
 
== Operations with 0 ==
Line 7: Line 11:
 
*If you subtract 0 from a number, the difference is that number. For example, <math>7-0=7</math>.
 
*If you subtract 0 from a number, the difference is that number. For example, <math>7-0=7</math>.
 
*If you subtract a number from 0, the difference is that number's opposite. For example, <math>0-3=-3</math>.
 
*If you subtract a number from 0, the difference is that number's opposite. For example, <math>0-3=-3</math>.
−
*If you multiply any amount of numbers by 0, the product is 0. For example, <math>3*6*0=0</math>.
+
*If you multiply any amount of numbers by any amount of 0's, the product is 0. For example, <math>3\times 0 \times 6\times 0=0</math>.
−
*You cannot divide a number by 0.
+
*A number divided by 0 is undefined.
 
*Dividing any number which is not equal to 0 will result in a quotient of 0. For example, <math>0\div 9=0</math>.
 
*Dividing any number which is not equal to 0 will result in a quotient of 0. For example, <math>0\div 9=0</math>.
 +
*There is a special case when you try to compute <math>0!</math>. The result is 1. Find more by clicking the 1 in brackets: [http://www.artofproblemsolving.com/Wiki/index.php/Factorial#Additional_Information].
 +
*<math>0^a=0</math> for any positive <math>a\in\mathbb R</math>.
 +
*<math>0^0</math> is <math>1</math> or undefined depending on the context.
 +
*<math>0</math> divided by <math>0</math> is indeterminate.
  
 
== See also ==
 
== See also ==
Line 17: Line 25:
  
 
[[Category:Constants]]
 
[[Category:Constants]]
−
 
+
[[Category:Integers]]
 
{{stub}}
 
{{stub}}

Latest revision as of 19:40, 6 October 2024

Zero, or 0, is the name traditionally given to the additive identity in number systems such as abelian groups, rings and fields (especially in the particular examples of the integers, rational numbers, real numbers and complex numbers).

The development of a concept and notation for 0, probably in ancient Indian civilization, and its subsequent transmission to Europe via the Persians and Arabs, was fundamental to the success of western mathematics in fields beyond geometry. It has suprisingly much relevance due to its significance in positional number systems. For instance, normal commercial interactions might be seriously slowed if cashiers had to make change on a purchase of LXIV dollars with bills marked L, X, V and I when handed XC dollars.

Why Teachers warn you not to divide by 0

Teachers keep saying, " Do not divide by zero." You might be thinking, Why? First, normally dividing by a smaller number gives you bigger results. So maybe $1/0$ equals ∞! But that is not the case. 1 over $x$ is the reciprocal or the multiplicative inverse of $x$. So anything times its reciprocal has to be 1. Therefore, 0 has no reciprocal because anything times 0 is 0.

Operations with 0

  • If you add a number to 0, the sum is that number. For example, $3+0=3$.
  • If you subtract 0 from a number, the difference is that number. For example, $7-0=7$.
  • If you subtract a number from 0, the difference is that number's opposite. For example, $0-3=-3$.
  • If you multiply any amount of numbers by any amount of 0's, the product is 0. For example, $3\times 0 \times 6\times 0=0$.
  • A number divided by 0 is undefined.
  • Dividing any number which is not equal to 0 will result in a quotient of 0. For example, $0\div 9=0$.
  • There is a special case when you try to compute $0!$. The result is 1. Find more by clicking the 1 in brackets: [1].
  • $0^a=0$ for any positive $a\in\mathbb R$.
  • $0^0$ is $1$ or undefined depending on the context.
  • $0$ divided by $0$ is indeterminate.

See also

This article is a stub. Help us out by expanding it.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Zero_(constant)&oldid=229172"