Proof
The proof is straightforward. Starting from the right-hand side, apply the distributive law to get
- ,
and set
as an application of the commutative law. The resulting identity is one of the most commonly used in mathematics.
The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring R.
Conversely, if this identity holds in a ring R for all pairs of elements a and b of the ring, then R is commutative. To see this, we apply the distributive law to the right-hand side of the original equation and get
and for this to be equal to, we must have
for all pairs a, b of elements of R, so the ring R is commutative.
Read more about this topic: Difference Of Two Squares
Famous quotes containing the word proof:
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (1819–1891)
“If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
—Susan B. Anthony (1820–1906)
“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)