Tikal-25 Manual - Special Functions

Functions taking on a single argument - as opposed to binary operations like +,-,× etc. - operate instantly on the currently displayed value and replace it with their result. They can be executed at any point in a calculation without interfereing with the priority of pending operations.

The square key x² calculates the square of the currently displayed number, while INV x² produces the square root √x instead.

Generic powers are calculated by the y^x key, generic roots by INV y^x providing ^x√y.

These functions are the only special functions using two arguments and will not display a result immediately when pressed. After entering the y argument and pressing y^x or INV y^x, the x argument must be entered and then = or another operation key pressed to display the result. If no x argument is given, y is reused as the second argument, providing y^y or ^y√y respectively.

The y value is restricted to non-negative numbers. When y is negative, the calculator enters its error condition and displays "E", so CLR has to be pressed to continue anew. Any non-negative number raised to 0 returns 1.

The reciprocal key 1/x returns for all x ≠ 0 the value 1/x, as could be expected.

The factorial key x! returns the factorial 1 x ... x n for integer values 0 ≤ n ≤ 69. By definition, 0! returns 1.

The percent key % converts the displayed value from a percentage to a matching argument for the given operational context.

When % is pressed after an arithmetic operation, the value produced is either a percentage of the previous number in case of addition or substraction, or a percentge fraction in case of multiplication or division:

Argument 1	Operation	Argument 2	%	Computed percentage	=	Result
1500	+	7	%	105	=	1605
1500	-	7	%	105	=	1395
1500	×	7	%	0.07	=	105
1500	÷	7	%	0.07	=	21428.571

The key ln calculates the natural logarithm (log_e) of the currently displayed number. INV ln provides the inverse natural logarithm of the currently displayed number x, which is e^x.

The key lg calculates the common logarithm (log₁₀) of the currently displayed number. INV lg computes the inverse common logarithm of the currently displayed number x, which is 10^x.

The key lb on level II calculates the binary logarithm (log₂) of the currently displayed number. INV lb computes the inverse binary logarithm of the currently displayed number x, which is 2^x.

The key log y on level II calculates the general logarithm (log_y) of the currently displayed number. INV log y computes the inverse general logarithm of the currently displayed number x to the base y, which is y^x. Contrary to the power function, here the first arg is the exponent and the second arg is the base, so the inverse function can be applied directly to the result of the general logarithm.

log_yx is computed as ln x / ln y.

The trigonometrics functions work internally in Radian, so entering angles in degrees may lead to somewhat higher rounding errors due to the conversion happening with the Decimal representation of π which has a high but finite accuracy.

The Degree, Radian, Grad key DRG rotates through the units for angular measurement in that same sequence. After starting the application, the default unit always is degree. Pressing the DRG once then switches to radian mode, once again to grad, and with a third press back to degree. The angular mode only has effect on the trigonometric functions. Since these are executed immediately, consecutive trigonometric functions within the same problem can be calculated in different angular modes according to requirements.

The accuracy and scope of the trigonometric functions depend on the underlying mathematic libray functions based on the Double type. In this version of the emulator no attempt is made to accurately emulate the microchip used in the original, not least for the reason that I do not have any detailed documentation about the inner workings of that chip.

To convert between angular units for a specific angle, use the follwing sequence:

• Press DRG once or twice until the source unit is displayed.
• enter the numeric value of the angle.
• Press SIN to calculate the sinus
• Press DRG once or twice until the source unit is displayed.
• Press INV SIN to calculate the arcus sinus in the desired unit format.

The returned values will be normalized to the unit circle.

Problems involving hyperbolic functions can also be solved using the exponential capability using INV LN instead of the shortcut level II keys.

Hyperbolic sine (sinh) x = 1/2 (e^x - e^-x) = (e^2x - 1)/(2e^x)

Hyperbolic cosine (cosh) x = 1/2 (e^x + e^-x) = (e^2x + 1)/(2e^x)

Hyperbolic tangent (tanh) x = (e^x - e^-x)/(e^x + e^-x) = (e^2x - 1)/(e^2x + 1)

sinh^-1 x = ln(x + √(x² + 1))

cosh^-1 x = ln(x + √(x² - 1)) for x ≥ 1

tanh^-1 x = 1/2 ln( (1 + x)/(1 - x) ) for -1 < x < 1

The 2nd rnd key produces random numbers in the range [0...1]. This is not part of the original capabilities, but may come in handy with statistical / stochastic simulations. To produce a sequence of random numbers in any closed range [0...n] where n > 0, use this:

2nd rnd x 2nd K n =

2nd rnd =

...