Example: **502** (five hundred & two) could be mistaken for **52** (fifty two) without the zero in the tens place.

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Zero is neither negative nor positive. But it is an even number.

## The Idea

The idea of **zero**, though natural to us now, was not natural lớn early humans ... If there is nothing to lớn count, how can we count it?

**Example: you can count dogs, but you can"t count an empty space:**

Zero Dogs? Zero Cats?

An empty patch of grass is just an empty patch of grass!

## Zero as a Placeholder

But about 3,000 years ago people needed khổng lồ tell the difference between numbers lượt thích **4** & **40.** Without the zero they look the same!

So zero is now used as a "placeholder": it shows "there is no number at this place", like this:

502 | This means 5 hundreds, |

## The Value of Zero

Then people started thinking of zero as an actual **number**.

### Example:

"I had 3 oranges, then I ate the 3 oranges, now I have **zero** oranges...!"

## Additive Identity

And zero has a special property: when we địa chỉ cửa hàng it to lớn a number we get that number back, unchanged

### Example:

7 + 0 = 7

Adding 0 to lớn 7 gives the answer 7

Also 0 + 7 = 7

This makes it the **Additive Identity**, which is just a special way of saying "add 0 và we get the **identical** (same) number we started with".

## Special Properties

Here are some of zero"s properties:

Property Example

a + 0 = a | 4 + 0 = 4 |

a − 0 = a | 4 − 0 = 4 |

a × 0 = 0 | 6 × 0 = 0 |

0 / a = 0 | 0/3 = 0 |

a / 0 = undefined (dividing by zero is undefined) | 7/0 = undefined |

0a = 0 (a is positive) | 04 = 0 |

00 = indeterminate | 00 = indeterminate |

0a = undefined (a is negative) | 0-2 = undefined |

0! = 1 ("!" is the factorial function) | 0! = 1 |

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### Zero number definition

Zero is a number used in mathematics khổng lồ describe no quantity ornull quantity.

When there are 2 apples on the table & we take the 2 apples, wecan say that there are zero apples on the table.

The zero number is not positive number & 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 & Arabs and later in Europe.

Symbol of zeroThe zero number is denoted with the0 symbol.

The Arabic numeral system uses the٠ symbol.

### Zero number properties

x represents any number.

OperationRule

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 khổng lồ the number:

x + 0 = x

For example:

5 + 0 = 5

Zero subtractionSubtraction of a number minus zero is equal khổng lồ the number:

x - 0 = x

For example:

5 - 0 = 5

Multiplication by zeroMultiplication of a number times zero is equal lớn 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 khổng lồ the zero powerThe power nguồn 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 lớn get zero.

Only the limit of the base b logarithm of x, when x convergeszero is minus infinity:

### Sets that contain zero

Zero is an element of the natural numbers, integer numbers, real numbersand complex numbers sets:

SetSet 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 phối 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 thành viên of the even numbers set:

0 ∈ 2k,k∈ℤ

So zero is an even number & not an odd number.

Is zero a natural number?There are two definitions for the natural numbers set.

The mix 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 thành viên of the set of non negative integers:

0 ∈ ℕ0

Zero is not a member of the phối of positive integers:

0 ∉ ℕ1

Is zero a whole number?There are three definitions for the whole numbers:

The mix 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 thành viên of the set of integer numbers và the set of nonnegative integers:

0 ∈ ℤ

0 ∈ ℕ0

Zero is not a thành viên of the phối of positive integers:

0 ∉ ℕ1

Is zero an integer number?The mix of integer numbers:

ℤ =0,1,2,3,4,5,6,7,8,...

Zero is a member of the phối 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 thequotient of two integer numbers:

ℚ =n/m; n,m∈ℤ

Zero can be written as a quotient of two integer numbers.

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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 thanzero: