Alternative Proof to show any integer multiplied with 0 is 0

In summary, the conversation discusses two different proofs for the property that ##a \cdot 0 = 0## for all ##a## using only basic properties of integers. The first proof uses the distributive property, while the second proof uses the property that ##a+0=a## for all integers. It is noted that the second proof is essentially the same as the one given in Spivak's book. The question of whether closure is necessary in the proof is also brought up.
  • #1
Seydlitz
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In his book, Spivak did the proof by using the distributive property of integer. I am wondering if this, I think, simpler proof will also work. I want to show that ##a \cdot 0 = 0## for all ##a## using only the very basic property (no negative multiplication yet).

For all ##a \in \mathbb{Z}##, ##a+0=a##.

We just multiply ##a## again to get ##a^2+(a \cdot 0) = a^2##. Then it follows ##a \cdot 0 = 0##. (I remove ##a^2## by adding the additive inverse of it on both side)
 
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  • #2
That is essentially the same proof as the one given in Spivak. I have no idea what simplification you think it affords.
 
  • #3
jgens said:
That is essentially the same proof as the one given in Spivak. I have no idea what simplification you think it affords.

I'm glad then that it's the same. Because I thought it's fallacious because I haven't showed if the integers are closed with multiplication, and Spivak's proof is the more appropriate one.
 
  • #4
Seydlitz said:
Because I thought it's fallacious because I haven't showed if the integers are closed with multiplication, and Spivak's proof is the more appropriate one.

Closure is not necessary in this argument.
 
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  • #5


Your proof is indeed correct and valid. It is a simple and elegant way to prove that any integer multiplied with 0 is 0. However, it is worth noting that Spivak's proof using the distributive property is also valid and perhaps more widely used because it can be extended to more complicated mathematical concepts. Both proofs demonstrate the fundamental property of multiplication with 0, which is that any number multiplied by 0 results in 0. This is an important concept in mathematics and is used in many different areas, such as algebra, calculus, and number theory. Your proof is a great example of how using basic properties can lead to a simple and effective proof.
 

1. What is an alternative proof for showing that any integer multiplied with 0 is 0?

The alternative proof for this concept uses the property of addition and multiplication known as the zero property. This property states that any number multiplied by 0 is equal to 0. Therefore, when an integer is multiplied by 0, the result will always be 0.

2. How is the alternative proof different from the traditional proof?

The traditional proof for this concept uses the definition of multiplication and the concept of identity elements to show that any number multiplied by 0 is 0. The alternative proof, on the other hand, uses the zero property, which is a more concise and direct approach.

3. Can the alternative proof be applied to all types of numbers?

Yes, the alternative proof can be applied to all types of numbers, including integers, fractions, and even complex numbers. This is because the zero property is a fundamental property of addition and multiplication, which holds true for all types of numbers.

4. Is the alternative proof considered a valid proof in mathematics?

Yes, the alternative proof is considered a valid proof in mathematics. It follows the logical reasoning and properties of addition and multiplication, making it a sound and accepted method for proving this concept.

5. How can the alternative proof be useful in solving mathematical problems?

The alternative proof can be useful in simplifying mathematical equations and solving problems involving multiplication. By using the zero property, we can easily eliminate terms and simplify expressions, making it a powerful tool in problem-solving.

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