# iCalcula: Real numbers

November 9, 2017

Real numbers are stored in the memory stack using only Part1 of the registers. Part2 remains null.

Following operators perform computations on real numbers:

|+| : Computes (y + x)-registers and stores result in x-register

|-| : Computes (y - x)-registers and stores result in x-register

|×| : Computes (y * x)-registers and stores result in x-register

|÷| : Computes (y / x)-registers and stores result in x-register

|1/x| : Computes inverse of x-register and stores result in x-register

|x²| : Computes square of x-register and stores result in x-register

|√x| : Computes square root of x-register and stores result in x-register

|y ˣ| : Computes power x-register of y-register and stores result in x-register

|ˣ√y| : Computes x-register root of y-register and stores result in x-register

|±| : Changes sign to x-register

Following groups of functions perform computations on real numbers:

• Exponential and logarithmic functions

• Parts functions

• Power functions

• Trigonometric functions

• Hyperbolic functions

• Percentage functions

• Conversion functions

• Combinatorics functions

• Statistic functions

• First and second kind of Bessel functions

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

|Exp| key enters Exponential menu with functions:

|e^x| : Natural exponential (base e)

|10^x| : Common exponential (antilogarithm)

|Log| key enters Logarithm menu with functions:

|ln(x)| : Natural logarithm (base e)

|log(x)| : Common logarithm (base 10)

PARTS FUNCTIONS

|Parts| key enters Parts menu with functions:

|Integral Part| : The fractional part is truncated

|Fractional Part| : Gives a number with null integral part

|Absolute Value| : For real numbers, |x-register|

|Argument Value| : Not used for real numbers

|Sign Value| : Gives +1 or -1 if number is > 0 or < 0

|Quotient of Division| : Quotient of y ÷ x - registers

|Reminder of Division| : Reminder of y ÷ x - registers

|Number to Fraction| : Converts a decimal to a pure fraction

|Prime Factors| : Calculates x register Prime Factors into a vector

TRIGONOMETRIC FUNCTIONS

|Trig| key enters Trigonometric Functions menu with functions:

|Sin| : Sin(x-register)

|Cos| : Cos(x-register)

|Tan| : Tan(x-register)

|ArcSin| : ArcSin(x-register)

|ArcCos| : ArcCos(x-register)

|ArcTan| : ArcTan(x-register)

The default angular mode specifies which unit of measure to assume for angles used in trigonometric functions. To set an angular mode enter |SYS| menu and press |Angles in Radians| or |Angles in Degrees|. Default angular mode system flag is displayed in yellow at the top of the display. Explicit angular mode through unit specification (adimensional subunit in |UNIT| menu) in a numeric input overrides default angular mode.

HYPERBOLIC FUNCTIONS

|Hyp| key enters Hyperbolic Functions menu with functions:

|Hyp Sin| : Hyperbolic Sin(x-register)

|Hyp Cos| : Hyperbolic Cos(x-register)

|Hyp Tan| : Hyperbolic Tan(x-register)

|Hyp ArcSin| : Hyperbolic ArcSin(x-register)

|Hyp ArcCos| : Hyperbolic ArcCos(x-register)

|Hyp ArcTan| : Hyperbolic ArcTan(x-register)

PERCENTAGE FUNCTIONS

|%| : x% of y

|% chg| : Percentage change from y to x

The percentage functions are special (compared with |×| and ÷) because they preserve the value of the base number (in the y-register) when they return the result of the percentage calculation (in the x-register). You can then carry out subsequent calculations using both the base number and the result without reentering the base number.

CONVERSION FUNCTIONS

|Conv| key enters Conversion Functions menu with functions:

|Degrees to Radians| : Converts an angle in x-register from degrees to radians

|Radians to Degrees| : Converts an angle in x-register from radians to degrees

COMBINATORICS AND RANDOM DISTRIBUTION FUNCTIONS

|Comb| key enters Combinatorics Functions menu with functions:

|Factorial| : Factorial of a non negative integer in x-register

|Gamma Function| : Gamma function of a non negative non integer in x-register

|n,m Binomial| : Computes the the binomial coefficient corresponding to the combinations of n elements (y-register) in subsets containing m elements (x-register)

|n Permutations| : Computes the permutations of n elements (x-register)

|k,n Permutations| : Computes the partial k-permutations (x-register) of n elements (y-register)

|k,n Permutations with rep| : Computes the partial k-permutations (x-register) with repetition of n elements (y-register)

|k,n Combinations| : Computes the k-combinations (x-register) of n elements (y-register)

|k,n Combinations with rep| : Computes the k-combinations (x-register) with repetition of n elements (y-register)

|[0,1] Uniform Random| : Generates a random number from a uniform distribution between 0 and 1 and stores it in x-register after lifting the stack

|{m0, s1} Gaussian Random| : Generates a random number from a normal distribution with mean 0 and standard deviation 1 and stores it in x-register after lifting the stack

|Dim X Uniform Random Vec| : Generates a vector with range given by x-register whose components are random numbers from a uniform distribution between 0 and 1 and stores it in x-register after lifting the stack

|Dim X Gaussian Random Vec| : Generates a vector with range given by x-register whose components are random numbers from a normal distribution with mean 0 and standard deviation 1 and stores it in x-register after lifting the stack

STATISTICS FUNCTIONS

The statistics menu provide functions to statistically analyze a set of one- or two-variable data (real numbers). Data values are accumulated as summation statistics in six statistics registers (Data Count, Sum x, Sum x², Sum y, Sum y² and Sum xy). It is necessary to clear the statistics register before entering a new set of statistical data.

Pressing |∑+| enters two variables into the statistics registers. The value already in the y-register is accumulated as a y-value. The value in the x-register is accumulated as a x-value. If x and y-registers contain real vectors, each of its elements are introduced into the statistics registers. If only x-register contain a real vector, y values are considered 0 as they are introduced in the statistics registers.

If you make a mistake when entering statistical data, reenter the mistaken x and y-values, press |∑-|, enter the correct pair of values and press |∑+| to accumulate them. The same is valid for vector values.

|Stat| key enters Statistics Functions menu with functions:

|Mean| : Computes mean of accumulated x and y-values and stores them in x and y-registers

|Weighted Mean| : Computes a weighted mean of accumulated x (data) and y (weights)-values and stores it in x-register

|Standard Deviation| : Computes population standard-deviation of accumulated x and y-values

|Sample Standard Deviation| : Computes sample standard-deviation of accumulated x and y-values

|Mean Deviation| : Computes the standard-deviation of the mean

|Mean Confidence| : Computes the semi-confidence interval for the mean. You can set confidence level to nσ with function |Set Confidence to sigma(x)| in |SYS| menu, where x is an integer in x-register

|LR Coefficients| : Computes Linear Regression, finding a straight line that best fits all the x and y-values. Coefficients of the straight line are stored in x (slope of the line) and y (y-intercept) - registers

|LR Deviations| : Computes standard-deviation of the coefficients of the linear regression

|LR Confidence| : Computes the semi-confidence interval for the coefficients of the linear regression

|Data Count| : Output the number of stored points in x-register

|Sum x| : Press to recall the sum of the x-values

|Sum x²| : Press to recall the sum of the squares of the x-values

|Sum y| : Press to recall the sum of the y-values

|Sum y²| : Press to recall the sum of the squares of the y-values

|Sum xy| : Press to recall the sum of the products of the x and y-values

|Stat Clear| : Clears the statistics registers

|Stat Data to Vector| : Stores all x-values as components of a vector in x-register and all y-values as componentes of a vector in y-register

|Linear Fit Chart| : Represents a scatter chart with (x,y) pairs of values and the corresponding linear regression

|Histogram Chart| : Represents an histogram chart with x-values

FIRST AND SECOND KIND OF BESSEL FUNCTIONS

|SP FNC| key enters Special Functions menu.

|BESSEL FUNCTIONS| enters submenu with functions:

|First Kind J0| : Computes Bessel Function of the first kind of order 0 for x-register value

|First Kind J1| : Computes Bessel Function of the first kind of order 1 for x-register value

|First Kind Jn| : Computes Bessel Function of the first kind of order n (y-register) for x-register value

|Second Kind Y0| : Computes Bessel Function of the second kind of order 0 for x-register value

|Second Kind Y1| : Computes Bessel Function of the second kind of order 1 for x-register value

|Second Kind Yn| : Computes Bessel Function of the second kind of order n (y-register) for x-register value

|COMPARISONS| enters submenu with functions:

|X = Y| : Computes 1 if x and y -registers are equal. Computes 0 otherwise.

|X > Y| : Computes 1 if x - register  >  y -register. Computes 0 otherwise.

|X >= Y| : Computes 1 if x - register  >=  y -register. Computes 0 otherwise.

|X < Y| : Computes 1 if x - register  <  y -register. Computes 0 otherwise.

|X <= Y| : Computes 1 if x - register  <=  y -register. Computes 0 otherwise.

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