  Built-in Functions (tap gold key twice for tertiary functions) LinEq2 Solves the system of linear equations in two variables: a11x + a12y = b1 a21x + a22y = b2 with a11, a12, b1 in registers R4, R5, R6 a21, a22, b2 in registers R1, R2, R3 Solutions are returned in X and Y, the determinant in Z. LinEq3 Solves the system of linear equations in three variables: a11x + a12y + a13z = b1 a21x + a22y + a23z = b2 a31x + a32y + a33z = b3 with a11, a12, a13 in registers R7, R8, R9 a21, a22, a23 in registers R4, R5, R6 a31, a32, a33 in registers R1, R2, R3 b1, b2, b3 in the extended registers R.9, R.6, R.3 Solutions are returned in X, Y, and Z, the determinant in T. REG / STK / SPC – Print Registers / Print Stack / Print Space These commands are only available if the printer is set to "Visible" (RPN-45 HD) in Settings, or if printer power is on (RPN-45 SD). They appear in place of the unit conversions after the gold key has been pressed twice. REG prints the basic registers 0...9, followed by extended registers 0...9 in a lighter color. In mode "CMPLX", complex registers 0...9 are shown, each with real and imaginary parts. STK prints the four stack registers. Note that – like on the HP-91 – a number entry immediately following STK doesn't lift the stack. SPC advances the paper tape by one line. Hold key down for auto-repeat. Quad Solves the quadratic equation ax2 + bx + c = 0, with a in Z, b in Y, c in X. The solution is returned in X and Y. If the result is a complex number, then X = real part, Y = imaginary part. Cube Solves the cubic equation ax3 + bx2 + cx + d = 0, with a in T, b in Z, c in Y, d in X. The real root is returned in X. The stack registers Y and Z receive the other two real roots (possibly identical) or – if complex conjugate – the real part (Y) and the imaginary part (Z). The type of roots is stored in stack register T as follows: 3: three real and distinct roots 2: three real roots of which at least two are equal 1: one real and two complex roots 0: not a cubic equation Tip: Use LAST STK to restore the equation on the stack. logy Computes logy(x), the logarithm of x to the base y, for arbitrary values (incl. complex). DATE Returns the date and the day of the week for the date in X. A whole number n in X returns the (Gregorian, if possible) date and day of the week n days from the date in Y, or from the current date if y is not a valid date. Special case: if Y = 0 and X between 1583 and 4099, the Easter date of that year is shown. ∆DYS Calculates in X the number of days between the dates in X and Y. Stack register Y receives the same number on the basis of a 30-day month. Dates must be entered as d.mmyyyy (U.S.A. and English-speaking Canada: m.ddyyyy). The range is Jan 1, 1 through Dec 31, 9999. Oct 15, 1582 and later dates are Gregorian, Julian if date is negative. INT / FRAC INT calculates the integer part, FRAC the fractional part of the display value. RAN# Generates a random number rn. Stack registers Y and Z determine the range. If both are zero, rn will be between 0 and less than 1. If both are non-consecutive integers, rn will be an integer in the range y through z. Otherwise, rn is non-integer in the specified range. RND Rounds the display value according to the fixed decimal point setting. L.R. – Linear Regression Calculates the linear least squares fitting line y = Ax + B of the data pairs accumulated by the summation function Σ+. Results are stored as follows:X = y-intercept B, Y = slope A, Z = correlation coefficient r, T = covariance qxy. x^,y^,r – Linear Estimation and Correlation Coefficient Using the value in X, calculates – based on linear regression of the accumulated data pairs – an estimated value for y (in X). In addition, Y receives an estimated value for x if the value in Y is assumed to be an y input value. The correlation coefficient r is stored in Z. xw,s – Weighted Mean Computes an average by giving different weights (or frequencies) to individual values. For each data pair, enter the value in X, the weight in Y. The mean is returned in X, the standard deviation in Y. Prob – Gaussian Probability Function Returns, for the standard normal distribution, the value of the probability density function φ(x) in X, and the value of the cumulative distribution function Φ(x) in Y. Example: φ(0) = 0.3989 (top of the bell-shaped curve), Φ(0) = 0.5 = 50% (area to the left) Py,x – Permutations Counts the number of ways to arrange y differents elements into sets of x elements. The order is important and repetitions or recurrence is not allowed. Example: Car race with 32 cars. How many ways for first 4 ranks? P32,4 = 863,040 Cy,x – Combinations Counts the number of ways to select x items out of a larger data set of y distinct items, where the order does not matter. Cy,x is called the binomial coefficient. Example: How many poker hands (5 cards from 52)? C52,5 = 2,598,560 – x –  – Print Display Value This command is only available if the printer is set to "Visible" (RPN-45 HD) in Settings, or if printer power is on (RPN-45 SD). It prints the currently displayed value on the paper tape. In mode "CMPLX", both real and imaginary parts of results will be printed. – Rip Off Paper Tape This command is only available if the printer is set to "Visible" (RPN-45 HD) in Settings. The paper tape will be removed almost completely. LAST x/STK – Last x and Last Stack STK restores the entire stack to the state before the last operation. Especially useful for functions returning more than one value. HEX – Hexadecimal Conversion shows the integer of the current display values as hexacecimal number. Negative values are shown as their 2's complement. Executing any key restores the standard display format. %∑ – Percent of Sum computes the percentage the display value is of a sum created by the summation key ∑+. Complex Numbers CPLX toggles complex number mode. Most functions and operations work just like with real numbers. All stack registers and the nine primary registers are available (including register arithmetic operations.) The polar-conversion key shows the angle used in phasor form. Entering a complex number: Real part ENTER imaginary part, followed by the key I Shortcut for a purely imaginary value: Imaginary part, then Re≷Im Shortcut for a purely real value: Real part, then next operation ClrIm clears the imaginary part of the display value. Re≷Im swaps the real and imaginary parts of the display value. STO  x≷y  n swaps the real and imaginary parts of register n (n = 1...9), or ofall registers (n = 0). Re↔Im swaps the real and imaginary parts of the display value. Conj changes the display value into its complex conjugate. Integral roots: yˣ for complex y stores in X and registers 1...9 up to 10 solutions if 1/x is a positive integer. Example 1:  (2 - 4√5 i):  2  ENTER  5  √ x  4  x  CHS  I Example 2:  2i * (-8+6i)3:  2  Re↔Im  8  CHS  ENTER  6  I   3   yx  x  =  -1872 + 704i Example 3:  i!:  1  Re≷Im  n!  =  0.49802 - 0.15495i  In portrait mode, tapping the register popover or the stack display below the display toggles between real and imaginary parts. Other Super-45 Enhancements FIX followed by the decimal point toggles thousand separators on and off. Access extended registers R1...R9 by tapping . or EEX after STO or RCL. In portrait mode, tap the register view to display the additional registers. Not available in complex mode. Gold-Gold  R↓ works as R↑. Gold-Gold  6 executes the ABS (absolute value) function. STO  R↓ copies all registers to the backup area. RCL  R↓ retrieves all registers from the backup area. STO (or RCL)  x≷y  R↓ swaps all registers with those in the backup area. STO  x≷y  n swaps regular and extended registers Rn (n = 1...9). In complex mode, this is equivalent to swapping the real and imaginary parts of register n. If n = 0, all 9 registers are swapped simultaneously. STO  x≷y  ∑+ swaps statistics registers R5...R8 with their extended counterparts. STO Σ+ may be used to directly store a 2-dimensional value (x to R7, y to R8). In complex mode, Σ+ and Σ- add/subtract the complex values in X and Y to R7 and R8. Register arithmetic also works with the "double register" Σ (R7/R8). RCL register arithmetic operations on register 0 use the LAST x value. Functions Σ+ and Σ- support the values Σy2, Σxy, and Σx2y in extended registers R6, R7 and R8. Functions Σ+ and Σ- report an error if the counter value in R5 is not a natural number. Last X after these functions retrieves both x and y values. Mean and standard deviation are calculated for both x and y values (if possible). Results for y are returned in Z (x̅) and T (s). The operations yx, logarithm and square root accept negative numbers. Arc sine and arc cosine accept input values smaller than -1 and larger than 1. The factorial operation n! is not restricted to natural numbers. Polar/Rectangular conversion works in hyperbolic mode HYP. Active prefix keys are visually marked. Prefix keys may be cancelled anytime by tapping them again. The current angular mode is underlined. Stack and storage registers are non-volatile. CLx, when executed as third function (CLR A), clears the stack and all registers, except those in the backup area. Small values which would show as zero will turn on SCI display mode automatically. Leading zeros are suppressed while entering a number. Storage register R9 is not cleared by trigonometric operations (black keys).