1 ENTER 3 / 3 * 1 -

[Dieter]

(01-26-2016 08:37 PM)Csaba Tizedes Wrote:  My favourite is 3^3-27, it's works well on those models which have integer exponentiation routine and do not use the a^b = EXP(b×LN(a)) method.

All HPs since the HP92 (IIRC) internally use additional guard digits so that 3^3 will return 27 (exactly) and 3^3–27 yields a plain zero. No integer exponentiation routine required.

The three guard digits are sufficient for most cases, yet not for all. Very large results > 10^20 may be off in the last digit. The 15C Advanced Functions Handbook states a possible error within 3 ULP. On calculators with a larger working range (x<10^500) the error may get somewhat larger. On the other hand, integer results within ±999999999[99] should be exact. Including 3^3. ;-)

(01-26-2016 08:37 PM)Csaba Tizedes Wrote:  Also interesting the (SQRT(2))^2-2 calculation.

Sqrt(2) is irrational, so there is no 10-digit (or 12-digit, or 16-digit...) value that exactly equals sqrt(2). The square of this result may happen to round to 2 or not.

On a 10-digit calculator sqrt(2) is returned as 1,414213562. This is the exact value for the true square root 1,4142435623730950488.. rounded to 10 digits.
1,414213562^2 again is 1,99999989447... which in turn is correctly displayed as 1,999999999. A calculator that returns a plain 2 for 1,414213562^2 is simply ...wrong. Simply because there is no 10-digit value which, when squared, rounds to 2 again. 1,414213562^2 yields 1,999999999 and 1,4142135623^2 returns 2,000000002.

Let us not forget: our calculators do not return sqrt(2), or sin(40°), or ln(3). They return a number that resembles the true result as closely as possible (within 10 or 12 digits). But this number is not identical with the true result.

(01-26-2016 08:37 PM)Csaba Tizedes Wrote:  And my favourite for SOLVE: 3×X+1÷(X-5) = 15+1÷(X-5) and solve it for X. (Hint: X=5 is wrong answer...)

My 35s returns 4,99999999999. ;-)

Which, in a way, is the "best" result you can get. For x=5+ε, the difference between the left and right hand side is 3 ε. Since the function is not defined for x=5, the value with the smallest possible ε comes as close to zero as possible.

Dieter