### Zero number definition

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 digit**

The 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 zero**

The zero number is denoted with the 0 symbol.

The Arabic numeral system uses the ٠ symbol.

### Zero number properties

*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 | |

**Zero addition**

Addition of a number plus zero is equal to the number:

*x + 0 = x*

For example:

*7 + 0 = 7*

**Zero subtraction**

Subtraction of a number minus zero is equal to the number:

*x – 0 = x*

For example:

*7 – 0 = 7*

**Multiplication by zero**

Multiplication of a number times zero is equal to zero:

*x × 0 = 0*

For example:

*7 × 0 = 0*

**Number divided by zero**

Division of a number by zero is not defined:

*x ÷ 0 is undefined*

For example:

*7 ÷ 0 is undefined*

**Zero divided by a number**

Division of a zero by a number is zero:

*0 ÷ x = 0*

For example:

*0 ÷ 7 = 0*

**Number to the zero power**

The power of a number raised by zero is one:

*x ^{0} = 1*

For example:

*7 ^{0} = 1*

**Logarithm of zero**

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:

### 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 ∈ ℕ^{0} |

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 4:

0 × 4 = 0

Zero is a member of the even numbers set:

0 ∈ {2*k*, *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:

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 ∈ 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 = 0/7

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:

7 > 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.