## July 17, 2008

### Too good to be true

(Question sent in by Udupendra. Thanks a lot amigo!)

Connect the equation (x + y)^{n} = x^{n} + y^{n} to these identities:

Cracked by Goyal , p vs np , Ajay Parasuraman , Rohan , insufferablejake , Tathagata Chatterjee , gammafunction , sidsen , malcaluffin , rajeshvj , Srivats and shenoyvarun86.

Answer:

Check out udupendra’s comment below for the answer straight from the horse’s mouth :-)

The equation is called a ‘freshman’s dream’ as such an elegant equation is too good to be true, and therefore is not. The two identities are called a ‘sophomore’s dream’ because of a similar too-good-to-be-true feel, but in this case considering that a year of study has gone in, it is actually true.

From Google:

Sophomore’s Dream

Aah.. the first identity is the “Freshman’s Dream”, which is a pretty bad approximation and which is an error “supposedly” made by freshmen.

The next set of identities is the “Sophomore’s Dream”, which was given by Leonard Euler’s protege, John Bernoulli, which is again, too-intuitive-to-be-true at first sight, but then, turns out to be actually valid!

Sophomore’s Dream

nice question!

the mistake (x+y)^n=x^n+y^n is commonly referred to as the “freshman’s dream”

while the definite integrals given are called the “sophomore’s dream”

Dai! Lord Binomial is going to whack your freshman ass, as (a+b)^2 = (a^2)+(b^2)+ 2ab (atleast thats what my high school maths book used to claim :P)

And so the first equation (which is wrong) is known as the ‘freshman’s dream’, while the latter identities are called the ‘Sophomore’s Dream’

Good question.

Sophomore’s Dream.

Regards,

Tathagata Chatterjee

The first equation is Freshman’s dream which is not true

the next two equations are known as the Sophomore’s dream which is actually true but has a too-good-to-be-true feel :-)

dreams of prospective scholars of varying degrees of youngness

Sophomore’s Dream discovered by Bernoulli.

The identities are referred to as Sophomore’s Dream => Something, too good to be true, but is in fact true.

The equation is referred to as Freshman’s Dream => The mistake, which was corrected by the Binomial Theorem.

The first, wrong, identity is the so-called ‘Freshmans Dream’, the correct result for which is given by the binomial theorem.

The next 2 identities are the Sophomores Dream, which similarly, are so simplistic, and yet, true..

Identity is known as the Freshman’s dream

Integrals are Sophomores dream

Quoting wiki both have

too-good-to-be-true feel… but only the latter is true