Operations With Zero

[nycmathguy]

1. Why is 0 - 0 = 0?

2. Why is 0 times any number 0?

3. Why is 0 + any number = any number?

4. Why is 0 times 0 = 0?

5. Why is 0! equal to 1?

 

[MathLover1]

The number 0 has been around to represent the idea of nothing since ancient Sumerian society, who used it to represent an absence of a number when writing out numbers and equations.
Zero is commonly used in language to express the concept of having none, and is used in math as an integer. The number 0 in today’s math can be tricky; why calculate something when there’s not actually anything there? But zero can be used in a variety of math problems, and it’s important to know what to do with zero when you see it.

1. Why is 0 - 0 = 0?
Identity Law of Subtraction states that zero subtracted from 0 is equal to itself.

2. Why is 0 times any number 0?

Multiplying by 0 is actually one of the easiest functions of 0. When you multiply by 0, the answer is always 0.

3. Why is 0 + any number = any number?

Identity Law of Addition states that any number added to 0 is equal to itself.
Like addition, if you subtract 0 from any number, you get the same sum. For example, 12-0 = 12

4. Why is 0 times 0 = 0?
same reason as in 2. When you multiply by 0, the answer is always 0.

5. Why is 0! equal to 1?

A factorial is a mathematical expression, expressed by ! that equals a number that is found by multiplying numbers all the numbers between 1 and the integer given.

So, 2! means we multiply all the numbers between 1 and 2. That means that 2! = 2×1 = 2 and therefore 2! = 2.
A zero factorial, often written as 0! Is defined as equal to 1.
Basically, since a factorial is an expression of the product of all the integers between the numbers given and 1, this is the only technically correct answer for 0! because the only number between 0 and 1 is 1.

few more:
One can argue that 0/0 is 0, because 0 divided by anything is 0.
Another one can argue that 0/0 is 1, because anything divided by itself is 1.
And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another.
To avoid "breaking math," we simply say that 0/0 is undetermined.

And, 0^0 = 1 =>The 0 exponent rule says that any base with an exponent of zero or 0 is equal to 1.
Meanwhile, 0 to any power equals 0. So 0² = 0.

 

[nycmathguy]

The number 0 has been around to represent the idea of nothing since ancient Sumerian society, who used it to represent an absence of a number when writing out numbers and equations.
Zero is commonly used in language to express the concept of having none, and is used in math as an integer. The number 0 in today’s math can be tricky; why calculate something when there’s not actually anything there? But zero can be used in a variety of math problems, and it’s important to know what to do with zero when you see it.

1. Why is 0 - 0 = 0?
Identity Law of Subtraction states that zero subtracted from 0 is equal to itself.

2. Why is 0 times any number 0?

Multiplying by 0 is actually one of the easiest functions of 0. When you multiply by 0, the answer is always 0.

3. Why is 0 + any number = any number?

Identity Law of Addition states that any number added to 0 is equal to itself.
Like addition, if you subtract 0 from any number, you get the same sum. For example, 12-0 = 12

4. Why is 0 times 0 = 0?
same reason as in 2. When you multiply by 0, the answer is always 0.

5. Why is 0! equal to 1?

A factorial is a mathematical expression, expressed by ! that equals a number that is found by multiplying numbers all the numbers between 1 and the integer given.

So, 2! means we multiply all the numbers between 1 and 2. That means that 2! = 2×1 = 2 and therefore 2! = 2.
A zero factorial, often written as 0! Is defined as equal to 1.
Basically, since a factorial is an expression of the product of all the integers between the numbers given and 1, this is the only technically correct answer for 0! because the only number between 0 and 1 is 1.

few more:
One can argue that 0/0 is 0, because 0 divided by anything is 0.
Another one can argue that 0/0 is 1, because anything divided by itself is 1.
And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another.
To avoid "breaking math," we simply say that 0/0 is undetermined.

And, 0^0 = 1 =>The 0 exponent rule says that any base with an exponent of zero or 0 is equal to 1.
Meanwhile, 0 to any power equals 0. So 0² = 0.

Perfect. Check out my thread zero^(zero).

 

[MathLover1]

proofs:

 

[nycmathguy]

proofs:

Thanks. I could have done that. I wanted your input. I will check out the clips.

 

[conway]

The number 0 has been around to represent the idea of nothing since ancient Sumerian society, who used it to represent an absence of a number when writing out numbers and equations.
Zero is commonly used in language to express the concept of having none, and is used in math as an integer. The number 0 in today’s math can be tricky; why calculate something when there’s not actually anything there? But zero can be used in a variety of math problems, and it’s important to know what to do with zero when you see it.

1. Why is 0 - 0 = 0?
Identity Law of Subtraction states that zero subtracted from 0 is equal to itself.

2. Why is 0 times any number 0?

Multiplying by 0 is actually one of the easiest functions of 0. When you multiply by 0, the answer is always 0.

3. Why is 0 + any number = any number?

Identity Law of Addition states that any number added to 0 is equal to itself.
Like addition, if you subtract 0 from any number, you get the same sum. For example, 12-0 = 12

4. Why is 0 times 0 = 0?
same reason as in 2. When you multiply by 0, the answer is always 0.

5. Why is 0! equal to 1?

A factorial is a mathematical expression, expressed by ! that equals a number that is found by multiplying numbers all the numbers between 1 and the integer given.

So, 2! means we multiply all the numbers between 1 and 2. That means that 2! = 2×1 = 2 and therefore 2! = 2.
A zero factorial, often written as 0! Is defined as equal to 1.
Basically, since a factorial is an expression of the product of all the integers between the numbers given and 1, this is the only technically correct answer for 0! because the only number between 0 and 1 is 1.

few more:
One can argue that 0/0 is 0, because 0 divided by anything is 0.
Another one can argue that 0/0 is 1, because anything divided by itself is 1.
And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another.
To avoid "breaking math," we simply say that 0/0 is undetermined.

And, 0^0 = 1 =>The 0 exponent rule says that any base with an exponent of zero or 0 is equal to 1.
Meanwhile, 0 to any power equals 0. So 0² = 0.

You can avoid breaking math, and solve for all expressions related to zero. Simply by allowing abstract numbers to contain abstract units. Just as concrete numbers contain concrete units.