Скачать или смотреть (4.8.37) Proof for the Additive Identity of Real Numbers

In this video, we aim to prove that there is at most one real number "a" such that "a + R" equals "R" for every real number "R." A number with this property is often referred to as an additive identity.

Let's start with assuming "a" is a real number. By subtracting "R" from both sides of the equation, we can simplify to "a" equals zero. Thus, we have identified one solution: "a" equals zero.

However, the task is to prove that there is at most one real number "a" which has the property that "a + R" equals "R" for all real numbers "R." We've identified one such solution, but how can we confirm that this is the only solution?

Imagine if we have a different real number "B" that isn't equal to "a" but also satisfies the equation for all real numbers "R." We can apply the same approach as we did with "a," subtracting "R" from both sides of the equation. Regardless of the value of "R," since "R" is a real number, we arrive at "B" equals zero.

But we know that zero is equivalent to "a," which implies "B" equals "a." The contradiction here is that "B" equals "a" and "B" does not equal "a" at the same time.

Everything we've done here is logically sound, yet our conclusion contradicts our assumption that "B" is both equal to and not equal to "a." This is an impossibility - "B" cannot simultaneously be equal to and not equal to "a." Therefore, our initial assumption must be incorrect.

Regardless of the value of "B", even if it's different from "a", we end up with a contradiction. Hence, "a" must be the only solution.

As we previously verified, "a" equals zero satisfies the condition of being an additive identity. Thus, "a" equals zero is the only solution that satisfies this property.

#AdditiveIdentity #MathProof #MathTutori