Ab=1/4((a+b)^2-(a-b)^2) Is An Amazing Identity, And Unfortunately,

Some People Think That Mathematics Is A Serious Business That Must Always Be Cold And Dry; But We Think Mathematics Is Fun, And We Aren't Ashamed To Admit The Fact. Why Should A Strict Boundary Line Be Drawn Between Work And Play? Concrete Mathematics Is Full Of Appealing Patterns; The Manipulations Are Not Always Easy, But The Answers Can Be Astonishingly Attractive.

A Lot Of The High-level Sports Are Really In Your Mind.

Ab=1/4((a+b)^2-(a-b)^2) Is An Amazing Identity, And Unfortunately, I Have To Remind My Current Students How To Prove It.

I Was Reminded Of The Sydney Harris Cartoon That Said 'adding Two Numbers That Have Not Been Added Before Does Not Constitute A Mathematical Breakthrough'.

Incidentally, When We're Faced With A "prove Or Disprove," We're Usually Better Off Trying First To Disprove With A Counterexample, For Two Reasons: A Disproof Is Potentially Easier (we Need Just One Counterexample); And Nitpicking Arouses Our Creative Juices. Even If The Given Assertion Is True, Our Search For A Counterexample Often Leads To A Proof, As Soon As We See Why A Counterexample Is Impossible. Besides, It's Healthy To Be Skeptical.