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 6th 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 
0^{5} = 0 
x ^{0} = 1 
5^{0} = 1 

Root 
√0 = 0 

Logarithm 
log_{b}(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:
x^{0} = 1
For example:
5^{0} = 1
The base b logarithm of zero is undefined:
log_{b}(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 ∈ ℕ_{0} 
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:
ℕ_{0} = {0,1,2,3,4,5,6,7,8,...}
The set of positive integers:
ℕ_{1} = {1,2,3,4,5,6,7,8,...}
Zero is a member of the set of non negative integers:
0 ∈ ℕ_{0}
Zero is not a member of the set of positive integers:
0 ∉ ℕ_{1}
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:
ℕ_{0} = {0,1,2,3,4,5,6,7,8,...}
The set of positive integers:
ℕ_{1} = {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 ∈ ℕ_{0}
Zero is not a member of the set of positive integers:
0 ∉ ℕ_{1}
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.
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