Zero is a number used in mathematics to describe no quantity or null quantity.
When there are 2 apples on the table and we take the 2 apples, we can say that there are zero apples on the table.
The zero number is not positive number and not negative number.
The zero is also a placeholder digit in other numbers (e.g: 40,103, 170).
Zero is a number. It is not positive nor negative number.
The zero digit is used as a placeholder when writing numbers.
For example:
204 = 2×100+0×10+4×1
The modern 0 symbol was invented in India in the 6-th century, used later by the Persians and Arabs and later in Europe.
The zero number is denoted with the 0 symbol.
The Arabic numeral system uses the ٠ symbol.
x represents any number.
| Operation | Rule | Example |
|---|---|---|
Addition |
x + 0 = x |
3 + 0 = 3 |
Subtraction |
x - 0 = x |
3 - 0 = 3 |
Multiplication |
x × 0 = 0 |
5 × 0 = 0 |
Division |
0 ÷ x = 0 , when x ≠ 0 |
0 ÷ 5 = 0 |
|
x ÷ 0 is undefined |
5 ÷ 0 is undefined |
|
Exponentiation |
0 x = 0 |
05 = 0 |
|
x 0 = 1 |
50 = 1 |
|
Root |
√0 = 0 |
|
Logarithm |
logb(0) is undefined |
|
 |
||
Factorial |
0! = 1 |
|
Sine |
sin 0º = 0 |
|
Cosine |
cos 0º = 1 |
|
Tangent |
tan 0º = 0 |
|
Derivative |
0' = 0 |
|
Integral |
∫ 0 dx = 0 + C |
|
 |
Addition of a number plus zero is equal to the number:
x + 0 = x
For example:
5 + 0 = 5
Subtraction of a number minus zero is equal to the number:
x - 0 = x
For example:
5 - 0 = 5
Multiplication of a number times zero is equal to zero:
x × 0 = 0
For example:
5 × 0 = 0
Division of a number by zero is not defined:
x ÷ 0 is undefined
For example:
5 ÷ 0 is undefined
Division of a zero by a number is zero:
0 ÷ x = 0
For example:
0 ÷ 5 = 0
The power of a number raised by zero is one:
x0 = 1
For example:
50 = 1
The base b logarithm of zero is undefined:
logb(0) is undefined
There is no number we can raise the base b with to get zero.
Only the limit of the base b logarithm of x, when x converges zero is minus infinity:
Zero is an element of the natural numbers, integer numbers, real numbers and complex numbers sets:
| Set | Set membership notation |
|---|---|
| Natural numbers (non negative) | 0 ∈ ℕ0 |
| Integer numbers | 0 ∈ ℤ |
| Real numbers | 0 ∈ ℝ |
| Complex numbers | 0 ∈ ℂ |
| Rational numbers | 0 ∈ ℚ |
The set of even numbers is:
{... ,-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, ...}
The set of odd numbers is:
{... ,-9, -7, -5, -3, -1, 1, 3, 5, 7, 9, ...}
Zero is an integer multiple of 2:
0 × 2 = 0
Zero is a member of the even numbers set:
0 ∈ {2k, k∈ℤ}
So zero is an even number and not an odd number.
There are two definitions for the natural numbers set.
The set of non negative integers:
ℕ0 = {0,1,2,3,4,5,6,7,8,...}
The set of positive integers:
ℕ1 = {1,2,3,4,5,6,7,8,...}
Zero is a member of the set of non negative integers:
0 ∈ ℕ0
Zero is not a member of the set of positive integers:
0 ∉ ℕ1
There are three definitions for the whole numbers:
The set of integer numbers:
ℤ = {0,1,2,3,4,5,6,7,8,...}
The set of non negative integers:
ℕ0 = {0,1,2,3,4,5,6,7,8,...}
The set of positive integers:
ℕ1 = {1,2,3,4,5,6,7,8,...}
Zero is a member of the set of integer numbers and the set of non negative integers:
0 ∈ ℤ
0 ∈ ℕ0
Zero is not a member of the set of positive integers:
0 ∉ ℕ1
The set of integer numbers:
ℤ = {0,1,2,3,4,5,6,7,8,...}
Zero is a member of the set of integer numbers:
0 ∈ ℤ
So zero is an integer number.
A rational number is a number that can be expressed as the quotient of two integer numbers:
ℚ = {n/m; n,m∈ℤ}
Zero can be written as a quotient of two integer numbers.
For example:
0 = 0/3
So zero is a rational number.
A positive number is defined as a number that is greater than zero:
x > 0
For example:
5 > 0
Since zero is not greater than zero, it is not a positive number.
The number 0 is not a prime number.
Zero is not a positive number and has infinite number of divisors.
The lowest prime number is 2.
