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).
Is zero a number?Zero is a number. It is not positive nor negative number.
Zero digitThe zero digit is used as a placeholder when writing numbers.
For example:
204 = 2×100+0×10+4×1
Zero number history Who invented the zero number?The modern 0 symbol was invented in India in the 6-th century, used later by the Persians and Arabs and later in Europe.
Symbol of zeroThe zero number is denoted with the 0 symbol.
The Arabic numeral system uses the ٠ symbol.
Zero number propertiesx represents any number.
Operation Rule Example Additionx + 0 = x
3 + 0 = 3
Subtractionx - 0 = x
3 - 0 = 3
Multiplicationx × 0 = 0
5 × 0 = 0
Division0 ÷ x = 0 , when x ≠ 0
0 ÷ 5 = 0
x ÷ 0 is undefined
5 ÷ 0 is undefined
Exponentiation0 x = 0
05 = 0
x 0 = 1
50 = 1
Root√0 = 0
Logarithmlogb(0) is undefined
Factorial0! = 1
Sinesin 0º = 0
Cosinecos 0º = 1
Tangenttan 0º = 0
Derivative0' = 0
Integral∫ 0 dx = 0 + C
Zero additionAddition of a number plus zero is equal to the number:
x + 0 = x
For example:
5 + 0 = 5
Zero subtractionSubtraction of a number minus zero is equal to the number:
x - 0 = x
For example:
5 - 0 = 5
Multiplication by zeroMultiplication of a number times zero is equal to zero:
x × 0 = 0
For example:
5 × 0 = 0
Number divided by zeroDivision of a number by zero is not defined:
x ÷ 0 is undefined
For example:
5 ÷ 0 is undefined
Zero divided by a numberDivision of a zero by a number is zero:
0 ÷ x = 0
For example:
0 ÷ 5 = 0
Number to the zero powerThe power of a number raised by zero is one:
x0 = 1
For example:
50 = 1
Logarithm of zeroThe 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:
Sets that contain zero
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 Is zero even or odd number?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.
Is zero a natural 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
Is zero a whole number?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
Is zero an integer number?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.
Is zero a rational 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.
Is zero a positive 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.
Is zero a prime 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.
See also e constant Factorial Logarithm Derivative