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:

Two Dogs
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, no tens, and 2 ones

## 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...!"

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

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 number history
Zero number properties
Sets that contain zero

### 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 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 & Arabs and later in Europe.

Symbol of zero

The zero number is denoted with the0 symbol.

The Arabic numeral system uses the٠ symbol.

### Zero number properties

x represents any number.

Operation
Rule
Example

x + 0 = x

3 + 0 = 3
Subtractionx - 0 = x3 - 0 = 3
Multiplicationx × 0 = 05 × 0 = 0
Division0 ÷ x = 0, when x ≠ 00 ÷ 5 = 0
x ÷ 0 is undefined5 ÷ 0 is undefined
Exponentiation0 x= 005= 0
x 0= 150 =1
Root√0 =0
Logarithmlogb(0)is undefined
Factorial0! = 1
Sinesin 0º = 0
Cosinecos 0º = 1
Tangenttan 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 subtraction

Subtraction of a number minus zero is equal khổng lồ the number:

x - 0 = x

For example:

5 - 0 = 5

Multiplication by zero

Multiplication of a number times zero is equal lớn zero:

x × 0 = 0

For example:

5 × 0 = 0

Number divided by zero

Division of a number by zero is not defined:

x ÷ 0 is undefined

For example:

5 ÷ 0 is undefined

Zero divided by a number

Division of a zero by a number is zero:

0 ÷ x = 0

For example:

0 ÷ 5 = 0

Number khổng lồ the zero power

The power nguồn of a number raised by zero is one:

x0 = 1

For example:

50 = 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 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:

Set
Set membership notation
Natural numbers (non negative)0 ∈ ℕ0
Integer numbers0 ∈ ℤ
Real numbers0 ∈ ℝ
Complex numbers0 ∈ ℂ
Rational numbers0 ∈ ℚ
Is zero even or odd number?

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: