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Acceleration: Acceleration felt by a mass m when acted on by a force F. This is Newton's Second Law of Motion. See also Force(). Accel( F, m ) = F/m  
Acos: Arccosine of x – Show the angle whose cosine is x (–1 ≤ x ≤ 1).  
Acosh: Arc hyperbolic cosine of x (x ≥ 1). Acosh(x) = 2*Ln(√ ½(x+1) + √ ½(x–1) )  
Aerobic Points Earned: Aerobic Points for 27 common workouts – based on Dr Kenneth Cooper's bestseller, Aerobics. The general objective is to get a minimum of 4 workouts per week and accumulate at least 30 points per week of aerobic exercise to gain a training effect. Approximate calories burned are also reported.  
Alternate Bases: Display N in base base. Both arguments are integers, (N < 2^{31}), and (2 ≤ base ≤ 36)  
Amdahl's Law: Amdahl's Law – the maximum theoretical speedup from using N processors on an application with pc percent parallelism (0 < pc < 100)  
Antilog: Inverse logarithm  the number whose logarithm is x  
Area Brick: Area of the rectangular solid with sides L, W, D (all > 0). AreaBrick(L,W,D) = 2(LW+WD+DL)  
Area Circle: Area of the circle with radius r > 0. AreaCircle(r) = πr²  
Area Circular Segment: Area of a circular segment of a circle with radius r and height h (both > 0).  
Area Cone: Area of a right cone of radius r and height h (both > 0). Area does not include the base. AreaCone(r,h) = πr√ r² + h²  
Area Cube: Area of a cube of side s, ( s > 0). AreaCube(s) = 6s²  
Area Cylinder: Area of a right cylinder of radius r and height h (both > 0). Area does not include base, top. AreaCylinder(r,h) = 2πrh  
Area Ellipse: Area of an ellipse with semiaxes of axis1 and axis2 (both > 0). AreaEllipse(a,b) = πab  
Area Frustum: Area of the curved surface of the frustum of a right cone with base radius b, top radius t, and height h (all > 0). AreaFrustum(b,t,h) = π(b+t)√ h²+(bt)²  
Area Lune: Area of a lune on the surface of a sphere of radius r included between two great circles whose inclination is x degrees: AreaLune(r,A) = 2πr²x/180  
Area Oblate Spheroid: Given an ellipse with axes axis1 and axis2, an oblate spheroid is formed by the rotation of the ellipse about its minor (shorter) axis. The surface area of an oblate spheroid is given by: AreaOblSpheroid(a,b) = π((2a²+(b²/E)*Ln((1+E)/(1E))) where (a > b), and E (Eccentricity) is: E = √ a²b² /a  
Area Parallelogram: Area of a parallelogram of sides side1, side2 and the angle angle (degrees) between them (all > 0): AreaPGram( s1, s2, r) = s1 × s2 × sin(2πr/180;)  
Area Prolate Spheroid: Given an ellipse with axes axis1 and axis2, a prolate spheroid is formed by the rotation of the ellipse about its major (longer) axis. The surface area of a prolate spheroid is given by: AreaProlSpheroid(a,b) = 2π(b²+(ab*Asin(E)/E)) where (a > b), and E (Eccentricity) is: E = √ a²  b² /a  
Area Pyramid: Lateral area of a regular pyramid of nSides sides, base dimension base (length of one side of base), and slant height height (all > 0). AreaPyramid(n,b,h) = ½nbh  
Area Rectangle: Area of a rectangle of sides length and width (both > 0). AreaRectangle(L,W) = LW  
Area Regular Polygon: Area of a regular polygon with n sides, each of length len > 0. See also RegPolygon() AreaRegPolygon(n,L) = nL²/(4tan(π/n))  
Area Spherical Cap: Area of a spherical cap of radius r and height h (both > 0). Note that r is the radius of the sphere, not the cap. A spherical cap is also known as a dome. See also AreaSphCap2(). AreaSphCap(r,h) = 2πrh  
Area Spherical Cap2: Area of a spherical cap of radius r and height h (both > 0). Note that r is the radius of the cap, not the sphere. A spherical cap is also known as a dome. See also AreaSphCap(). AreaSphCap2(r,h) = π(r² + h²)  
Area Sphere: Area of a sphere of radius radius > 0. AreaSphere(r) = 4πr²  
Area Spherical Polygon: Area of a spherical polygon of nSides sides, where angleSum is sum of its angles (degrees), and radius is the sphere's radius: AreaSphPolygon(r,n,S) = πr²((S/180)(n2)))  
Area Spherical Triangle: Area of spherical triangle whose angles are a1, a2, a3 (degrees) on a sphere of radius r (all > 0). 180 < (a1 + a2 + a3) < 540 AreaSphTriangle(r,A,B,C) = π(((A+B+C)/180)1)r²  
Area Square: Area of a square of side side > 0. AreaSquare(s) = s²  
Area Triangle: Area of a triangle of base base and height height (both > 0). AreaTriangle(b,h) = ½bh  
Area Triangle2: Area of a triangle given the coordinates of its 3 vertices (x1,x2), (y1,y2), (x3,y3). AreaTriangle2(x1,y1, x2,y2, x3,y3) = ½x1(y2y3)+x2(y3y1)+x3(y2y1)  
Area Triangle3: Area of a triangle given the lengths of its 3 sides a, b, c (all > 0). Using Heron's formula, let p = half perimeter = ½(a+b+c), then AreaTriangle3(a,b,c) = √p(pa)(pb)(pc)  
Arithmetic Last: The last element in the Arithmetic Progression defined by its first term (term1), delta (delta), and number of elements (numb) ArithmeticLast(a,d,n) = a+(n1)d  
Arithmetic Sum: The sum of the elements in the Arithmetic Progression defined by its first term (term1), delta (delta), and number of elements (numb) ArithmeticSum(a,d,n) = ½n(2a+d(n1)) Note that if the last term is L, the sum may be computed as: ArithmeticSum(a,n,L) = ½n(a+L); This program uses the first equation.  
Asin: Arcsine of x – Show the angle whose sine is x (–1 ≤ x ≤ 1).  
Asinh: Arc hyperbolic sine of x Asinh(x) = Ln(x + √ 1+x² )  
Atan: Arctangent of x – show the angle whose tangent is x  
Atan2: Arctangent of x/y, where the arguments are not both zero. Atan2() overcomes a limitation with Atan(). Specifically, with two arguments, Atan2() can return the angle in the correct quadrant  
Atanh: Arc hyperbolic tangent of x, (x < 1). Atanh(x) = ½(Ln(1+x) – Ln(1–x))  
 
Base: Display the integer argument N in octal, hex and binary  
Bill Split: The (rounded) amount each of N people should pay towards a total bill of (bill + tip), where tip is pc percent of bill. Note that (N > 1) and need not be an integer, and (0 ≤ pc ≤ 100). See also Tip().  
Binary: Display the integer argument N in binary  
Binomial: Display the coefficient of the x^{k} term of the expansion of (1 + x)^{n}. See also BinomCoefs(), BinomCum(), BinomProb(). Binom(n,k) = n!/(k!(nk)!)  
Binomial Coefficients: Display the (n+1) binomial coefficients of the expansion of (1 + x)^{n} for (n = integer) and (20 ≤ n < 0). See also Binom(), BinomCum(), BinomProb().  
Binomial: Cumulative Prob: Display the probability of achieving at most k successes in n trials, with p being the probability of success in one trial. The trials are independent, and the outcome of each is either success or failure. See also Binom(), BinomCoefs(), BinomProb(). BinomCum(p,n,k) = ∑_{x=0,k}{B(n,x)p^{x}q^{nx}} = ∑_{x=0,k}BinomProb(p,n,x) where (n,k = integers), (51 ≤ n < k ≤ 0), (0 < p < 1), q = (1p), and B(n,k) = n!/(k!(nk)!)  
Binomial Probability: Display the probability of achieving exactly k successes in n trials, with p being the probability of success in one trial. The trials are independent, and the outcome of each is either success or failure. The Binomial Probability Function is also known as the Probability Mass Function. See also Binom(), BinomCoefs(), BinomCum(). BinomProb(p,n,k) = B(n,k)p^{k}q^{nk} where (n,k = integers), (51 ≤ n < k ≤ 0), (0 < p < 1), q = (1p), and B(n,k) = n!/(k!(nk)!)  
Bit Count: Return the number of 1 bits in the 32bit integer argument N.  
Body Mass Index (BMI): BMI is a measure of weight for height in adults. As BMI increases, risk for many weightrelated diseases increases.  
 
Calendar: Display a calendar for year in (1753 – 9999), otherwise for the current year.  
Calories Burned: Approximate calories burned by a variety of common workouts and activities, including:
 
Capacitance: Capacitance for 210 capacitors (c1, c2, ...) in serial, parallel. All args > 0.  
Capacitance (Electrical): Electrical capacitance is the ability of a body to store electrical charge. It is the ratio of the change in electrical charge Q (coulumbs) in a system to a given electric potential V (volts). Electrical capacitance is measured in farads. CapElec( Q, V ) = Q/V  
Cartesian: The (x,y) coordinates of the point defined as dist units from the origin at the angle angle radians (that is, the arguments are the polar coordinates of the point)  
Celsius: Convert Fahrenheit temperature (x) to Celsius. Same as Centigrade(). See also Kelvin(). Celsius(x) = (5/9)*(x32) ≈ 0.5555556*(x32)  
Centigrade: Convert Fahrenheit temperature (x) to Centigrade. Same as Celsius(). See also Kelvin(). Centigrade(x) = (5/9)*(x32) ≈ 0.5555556*(x32)  
Circle3: Given 3 points on a plane, determine the radius and center of the the circle they define. The arguments are the 3 points.  
Circumference Circle: Circumference of a circle of radius r. CircumCircle(r) = 2πr  
Circumference Ellipse: Approximate circumference of an ellipse with semiaxes axis1 and axis2. CircumEllipse(a,b) ≅ 2π√ ½(a² + b²) is a good approximation, but a slightly better estimating function (from Ramanujan) is actually used.  
Coin Flip: Simulate a random coin flip.  
Colors: Display the color, the Red, Green, and Blue components of the colors (max=16) in a table, and show the HSV representation of each. Color is an integer representation of the color (e.g., 0x3DB733), and is silently kept in the range (0  0xFFFFFF). The values are sorted and duplicate values are silently ignored. See also RgbDisplay(), HSVtoRGB()  
Combinations: The number of combinations of n things taken m at a time. Both arguments are integers, and (n > m). Combinations(n,m) = n!/(m! (nm)!)  
Complex Add: Addition of complex numbers (x, y): where x = (a + bi), and y = (c + di) ComplexAdd(x, y) = (a+bi) + (c+di) = ((a+c) + (b+d)i)  
Complex Division: Division of complex numbers (x, y): where x = (a + bi), and y = (c + di) ≠ 0 ComplexDiv(x, y) = (a+bi)/(c+di)  
Complex Exp: Exponential of a complex number (x): where x = (a + bi) ComplexExp(x) = e^{(a+bi)}  
Complex Ln: Natural logarithm of a complex number (x): where x = (a + bi) ComplexLn(x) = ln(a+bi)  
Complex Multiply: Product of two complex numbers (x, y): where x = (a + bi), and y = (c + di) ComplexMult(x, y) = (a+bi) × (c+di) = ( (ac–bd) + (bc+ad)i)  
Complex Power: Raise a complex number (x) to a real power (s): where x = (a + bi) ComplexPower(x, s) = (a+bi)^{s}  
Complex Power2: Raise a complex number (x) to a complex power (y): where x = (a + bi), and y = (c + di) ComplexPower2(x, y) = (a+bi)^{(c+di)}  
Complex Roots: The n roots of a complex number (x): where x = (a + bi), and n is an integer in (212). ComplexRoots(x, n) = ^{n}√(a + bi)  
Complex Sqrt: The positive square root of a complex number(x): where x = (a + bi). When the result is (r + si), the other root is (–r – si). Note that r ≥ 0 (always) ComplexSqrt(x) = √(a + bi)  
Cos: Cosine of the angle x  
Cosh: Hyperbolic cosine of the angle x Cosh(x) = ½(e^{x} + e^{–x})  
Cross Product: Show the cross product of the two input 3D vectors (v1, v2) and the angle between them. The cross product is a vector perpendicular to both input vectors and normal to the plane defined by them. See also DotProd().  
Cube Root: Cube root of x  
Cubic Equation: The 3 solutions to the Cubic Equation: ax³ + bx² + cx + d = 0, (a ≠ 0)  
 
Date/Timer: Show the current date and time, and the elapsed time since the previous call (stopwatch capbility)  
DayofWeek: Print dayofweek for input date. See also DayOfYear(), DayOfYear2().  
DayofYear: Print dayofyear for input date. See also DayOfWeek(), DayOfYear2().  
DayofYear2: Print the date (dayofweek, month, day) for the N^{th} day of year year, (175 ≤ year ≤ 9999), (1 ≤ Numb ≤ 366). See also DayofYear(), DayOfWeek().  
Days Away: Display the date of the day which is nDays from date. nDays may be positive or negative. Note that the computation of years/days goes in the direction of nDays. That is, DaysAway( 740, 1994,FEB,22 ) will yield: Tuesday, February 22, 1994 → Sunday, March 3, 1996 = 740 days = 2 years, 10 days, but DaysAway( 740, 1996, MAR, 3 ) will yield: Tuesday, February 22, 1994 ← Sunday, March 3, 1996 = 740 days = 2 years, 9 days. See also DaysBetw().  
Days Between: Display the number of days between date1 and date2. Note that the computation of years/days goes from the first date to the second date. That is, DaysBetw( 2007, FEB, 15, 2008, MAR, 25 ) will yield: Thursday, February 15, 2007 → Tuesday, March 25, 2008 = 404 days = 1 year, 39 days, but DaysBetw( 2008, MAR, 25, 2007, FEB, 15 ) will yield: Thursday, February 15, 2007 ← Tuesday, March 25, 2008 = 404 days = 1 year, 38 days. See also DaysAway().  
Days Since: Display number of days since the last occurrence of the specified holiday. Holiday may be one of { CHRISTMAS, THANKSGIVING, VALENTINES, MOTHERS, FATHERS, SUMMER, FALL, WINTER, SPRING, LABOR, MEMORIAL }, among others. See also DaysTil(), DaysAway(), DaysBetw().  
Days Til: Display number of days til the next occurrence of the specified holiday. Holiday may be one of { CHRISTMAS, THANKSGIVING, VALENTINES, MOTHERS, FATHERS, SUMMER, FALL, WINTER, SPRING, LABOR, MEMORIAL }, among others. See also DaysSince(), DaysAway(), DaysBetw().  
Degrees: Convert r (radians) to degrees. See also Radians(): Degrees(r) = 180r/π = r × RADIANS2D ≈ r × 57.29578  
Density: The ratio of mass m to volume v Density( m, v ) = m/v  
Dice Toss: Simulate the random tossing of N dice. (1 ≤ N ≤ 6).  
Distance: Given two data points in 2D (or 3D) space, (x1, y1 [, z1]) and (x2, y2 [, z2]), compute the distance between them. When only 4 arguments are present, do the 2D computation. See also Line(), Line3D().  
Distance PointLine: Minimum distance between a point (x1, y1, [z1]) and a line denoted by its end points (x2, y2, [z2,]) and (x3, y3[, z3]). 2D or 3D: for 2D, the input points are two arguments each. See also LinePoint() and DistPtPlane().  
Distance PointPlane: Minimum distance between a point (x, y, z) and a plane represented by the equation: "Ax + By + Cz + D = 0." At least 1 of {A,B,C} must be nonzero. See also PlanePoint() and DistPtLine()  
Dot Product: Show the dot product of the 2 input vectors, each of length len. The dot product is a scalar value also known as the inner product or the scalar product. See also DotProd3d(), DotProdID()  
Dot Product 3D: Show the dot product of the 2 input 3D vectors (v1, v2) and the angle between them. The dot product is a scalar value also known as the inner product or the scalar product. See also CrossProd().  
Dot Product (Ids): Show the dot product of the two input vectors (ID={id1,id2}). The dot product is a scalar value also known as the inner product or the scalar product. See also DotProd(), CrossProd().  
 
Easter Date: Easter for year = year > 1752 (Easter falls between March 22 and April 25)  
Effective Rate: The effective annual rate for a nominal interest rate rate, compounded N times per year. rate must be > 0, and N is an integer > 1. The 'nominal rate' is the rate per compound period. For example, if a credit card charges a nominal 1.5%/month, the effective annual rate is 19.5618% (not 18%). See also NominalRate().  
Electrostatic Force: Coulomb's Law: The electrostatic force between two point charges (q1, q2) is proportional to their magnitudes and inversely proportional to the square of the distance d between them. Like charges repel, unlike attract. Similar to the Gravitational Force, except that the Gravitational Force always attracts. See also GravForce(). ElecForce(q1,q2,d) = k_{e}(q1 × q2)/d^{2}  
E = mc²: The rest energy (joules) equivalent to the mass mass (g). Energy computed using Einstein's famous equation: e = mc². For example, 1 gram ≈ 89.9×10^{12} joules ≈ 21.5 kilotons of TNT  
Erf: Error function of x  
Erfc: Complementary error function of x Erfc(x) = 1 – Erf(x)  
Exp: Exponential of x Exp(x) = e^{x}  
Exp: Exponential of (x – 1); accurate even for very small values of x Expm1(x) = e^{x}1  
 
Factor: List all the prime values which divide evenly (with no remainder) into integer N. Max value for N is (2^{31}1) = 2,147,483,647  
Factorial: The product of all the integers from 1 to integer N, where N must be in the range (0143). Note that by definition, Factorial(0) = 0! = 1. Factorial(n) = n! = 1×2×3×...×n  
Fahrenheit: Convert Centigrade (Celsius) temperature x to Fahrenheit. See also Rankine(). Fahrenheit(x) = 1.8x + 32  
Fibonacci: The N^{th} element in the Fibonacci series {Fibonacci(n), n = 0,1,2,3,...}. Fibonacci(0) = 0, Fibonacci(1) = 1, Fibonacci(n) = Fibonacci(n2)+Fibonacci(n1)  
Fitness Tests: Determine fitness level by taking one of six basic fitness tests (based on Dr Kenneth Cooper's Aerobics books). The tests are either Distance traveled in a specified time or Time to cover a specified distance for walking, running, swimming or cycling. Result is one of {VeryPoor, Poor, Fair, Good, Excellent, Superior}. Consult a physician before taking any of these tests. Warm up appropriately before starting a test.  
Force: The net force (newtons) on an object of mass m (kg) which induces an acceleration of a (m/s²). This is Newton's Second Law of Motion: F = ma  
Free Fall: Distance traveled in free fall (starting at rest) in time seconds. Distance is feet for Imperial, and meters for Metric. FreeFall( t ) = ½gt² where g is the appropriate standard gravity  
Future Value: The amount to which principal prin will accumulate in yrs years at an annual interest rate of rate percent compounded N times per year. If N is zero, simple interest is used. yrs and N are integers. See also PresentValue(), Ira(), IraProfile().  
 
GCD: Greatest common divisor (also known as greatest common factor (gcf)). The largest positive integer that divides the 32bit positive integer arguments evenly (no remainder). Special case: GCD(0,0) = 0  
Geometric Last: The last element in the Geometric Progression defined by its first term (term1), ratio (ratio), and number of elements (N) GeometricLast(a,r,n) = ar^{(n1)}  
Geometric Mean: Geometric mean of the arguments (all > 0). Not related to GeometricSum(), GeometricLast() M_{G} = (X_{1} × X_{2} × ... X_{n})^{1/n}  
Geometric Sum: The sum of the elements in the Geometric Progression defined by its first term (term1), ratio (ratio), and number of elements (N). N is an integer > 1, and ratio ≠ one or zero. 1: GeometricSum(a,r,n) = a(r^{n}1)/(r1)) Note that if the last term is L, the sum may be computed as: 2: GeometricSum(a,r,L) = (Lra)/(r1) Further, for n = ∞ and r² < 1.0; 3: GeometricSum(a,r) = a(1r) This program uses only Eq 1 above  
Gravitational Force: Newton's Law of Universal Gravitation: The force between two masses (m1, m2) is proportional to their magnitudes and inversely proportional to the square of the distance d between them. The masses are in kgs and the distance is in meters, so the force is in newtons. The force is always attractive. Similar to the Electrostatic Force, except that the Electrostatic Force may attract or repel. See also ElecForce(). GravForce(m1,m2,d) = G(m1 × m2)/d^{2}.  
GSeries: The N^{th} element in the Generalized Fibonacci Series {Gseries(n), n = 0,1,2,3,...}, defined as: GSeries(0) = v1, GSeries(1) = v2, GSeries(n) = GSeries(n2)+GSeries(n1)  
 
Harmonic Mean: Harmonic mean of the arguments (all arguments > 0). The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. M_{H} = n / (1/X_{1} + 1/X_{2} + ... + 1/X_{n})  
Heart Rate: Display maximum and various other workoutrelated target heart rates: age is age (integer, (10100)), and rPulse is resting heart rate (integer, (3590)).  
Heat Index: Display the heat index for a given temperature temp and relative humidity hum (%). The heat index is an approximation of the temperature humans perceive.  
Hex: Display the integer argument N in hexadecimal  
Hill Cipher: Encipher or decipher a message using the Hill Cryptosystem. User must have precreated an appropriate matrix and modular inverse matrix pair. See MatrixSave() and MatrixShow() to create and review a matrix set (you may have up to 20 sets, each based on one of three alphabets – sizes are {31,47,73} characters). Each matrix set has an alphabet id (13) associated with it at creation: • ID=1, 31 characters: AZ ., =# caseinsensitive, includes blank • ID=2, 47 characters: AZ 09 +*/ =().,# caseinsensitive, includes blank • ID=3, 73 characters: AZ az 09 +*/ =().,# casesensitive, includes blank To decipher a message, you must use the same matrix which enciphered it. Message size is limited to 140 characters.  
Holiday: List the holidays in month month. If month is not in JAN – DEC (1–12), list all the holidays.  
Horizon: Approximate distance to the horizon (on Earth) from a height of height (height > 0). In the Imperial System, distance is in miles and height is in feet. In the Metric System, distance is in kilometers and height is in meters.  
Horizon2: Approximate distance to the horizon on a sphere of radius radius from a height of height (height > 0). In the Imperial System, distance and radius are in miles and height is in feet. In the Metric System, distance and radius are in kilometers and height is in meters.  
HSVtoRGB: Given an input HSV (Hue, Saturation, Value) color representation, display its RGB (Red, Green, Blue) representation in a table. Hue is in degrees (0360), sat (saturation) and value are percents (1100), and all three arguments are silently kept in range. Value is also referred to as Brightness or Intensity.  
 
Integrate: _{a}∫^{ b}{expr(x)} dx. Approximate result of integrating the expression (expr) from a to b. A and b are the lower and upper bounds of the (dummy) variable in expr, and must be constants with (a ≠ b). The expression expr must have exactly one variable, must be singlevalued, and may be a single userdefined function or an expression with no userdefined functions. The expression expr must be continuous over the specified range, and may include any singlevalued function except {Rand(), IRand(), CoinFlip(), RockPS(), RockSpock(), Dice()}. Unpredictable results may occur when expr is not continuous or is otherwise illbehaved. Same as Quadrature(), except Quadrature() always shows convergence of the approximate solution. Quadrature() is strongly recommended.  
Intersect: Circle/Circle: Show the intersection points of the two input circles on a plane.  
Intersect: Circle/Line: Show the intersection points of a circle and a line on a plane.  
Intersect: Line/Line: Show the intersection point of a two lines on a plane.  
Intersect: Line/Plane: Show the intersection point of a line and a plane.  
Intersect: Line/Sphere: Show the intersection points of a line and a sphere.  
Intersect: Line/Triangle: Show the intersection points of a line and a triangle on a plane.  
Intersect: Plane/Plane: Show the equation of the line of intersection of two planes and the angle between them. The equation of a plane is Ax + By + Cz + D = 0.  
Intersect: Plane/Sphere: Show the intersection of a plane and a sphere. The equation of a plane is Ax + By + Cz + D = 0. The intersection will be a circle on the plane.  
Intersect: Sphere/Sphere: Show the intersection of two spheres. The intersection will be a circle on the plane perpendicular to a line connecting the centers of the spheres.  
Intersect: Two 2D Triangles: Show the intersection points of the two input triangles on a plane. There will be (06) points of intersection.  
Intersect: Two 3D Triangles: Show the intersection points of the two triangles in 3D space. If the triangles intersect, the result will be a line segment (part of the line of intersection of the two planes defined by the triangles).  
Inverse: Inverse of argument x (x ≠ 0) Inv(v) = 1/v  
InvMod: The modular multiplicative inverse of x mod y (Mod(x,y)), or zero if x and y are not coprime. A and b must both be integers > 1.  
IRA: The value of an investment (e.g., an IRA) in the future. value is the current value, add is the yearly addition to (or withdrawal from) the investment, rate is the annual interest rate (e.g., 5.25), and yrs is the duration (years). Ira() assumes Jan 1 investments and reports results for Dec 31. See also IraProfile(), FutureValue()  
IRand: Return a random integer value between integers m and n. Notes: (1) not reproducible, (2) IRand(0,1) is like a coin flip. See also Rand(), CoinFlip(), RockPS(), Dice().  
IRA Profile: Show lifetime investment progress (e.g., an IRA). value is the current value, add is the yearly addition to (or withdrawal from) the investment, rate is the annual interest rate (e.g., 5.25), and yrs is the lifetime (years). IraProfile() assumes Jan 1 investments and reports results for each Dec 31. See also Ira(), FutureValue()  
 
Kelvin: Convert Celsius temperature (x) to Kelvin. See also Rankine(). Kelvin(x) = x + 273.15  
Kinetic Energy: Kinetic Energy of an object of mass mass moving at a constant velocity vel. When mass is in kg and vel is m/sec, KEnergy is J (joules); when mass is in lb and vel is ft/sec, KEnergy is in ftlbf. See also PEnergy(). KEnergy(m,v) = ½mv²  
 
LCM: Least common multiple (also, lowest common factor (lcf)). The smallest positive integer that is a multiple of the 32bit positive integer arguments. Special case: if any argument is zero, the LCM is defined to be zero.  
Leap Year: Return 1 if year is a leap year, 0 otherwise. (1753 ≤ year ≤ 9999).  
Line: Given two data points on the XY plane, (x1, y1) and (x2, y2), compute the slope (m), the yintercept (b), the distance between the two points, the midpoint, and show the equation of the line containing the points. Same as Ymxb(). See also Line3D().  
Line (3D): Given two data points in 3D space (XYZ), (x1, y1, z1) and (x2, y2, z2), compute the equation of the line, the distance between the 2 points, the direction vector from point1 to point2, the midpoint, and the XY, XZ, YZ slopes. See also Line().  
Linear Equation: Solution to the linear equation: ax + b = 0 (a ≠ 0)  
Line Point: Minimum distance between a point (x1, y1, [z1]) and a line denoted by its end points (x2, y2, [z2,]), (x3, y3[, z3]) Also, show the direction vector and the closest point. 2D or 3D: for 2D, the input points are two arguments each. See also DistPtLine() and DistPtPlane(). Example: 2D distance from (5,7) to the line defined by (3,6) and (1,3), enter: 5,7, 3,6, 1,3  
Ln: Natural logarithm of x, (x > 0). The power to which e (2.718281828459...) must be raised to yield x  
Ln1p: (x > −1). Natural logarithm of (x+1). See Ln()  
Log: Logarithm (base 10) of x, (x > 0). The power to which 10 must be raised to yield x  
Log2: Logarithm (base 2) of x, (x > 0). The power to which 2 must be raised to yield x  
Lucas: The N^{th} element in the Lucas series {Lucas(n), n = 0,1,2,3,...}. Lucas(0) = 2, Lucas(1) = 1, Lucas(n) = Lucas(n2)+Lucas(n1)  
 
Math 101: A collection of 15 commonlyoccurring mathematical functions (e.g., √x , ln(x), e^{x}, ³√x ). Enter one argument and get all 15 results, as well as some basic mathematical facts.  
Matrix Copy: Copy the data for saved matrix (Id=inId) to matrix (Id=newId). (newId ≠ inId). The id's are in (120). See also MatrixDel(), MatrixSave(), MatrixShow().  
Matrix Delete: Delete Matrix (ID=mtxId). See also MatrixSave(), MatrixShow().  
Matrix Determinant: Compute the determinant of the input N×N matrix. N is in {2,3,4}. The N² elements of the matrix are in the mtx argument, rowwise and separated by commas. See also MatrixInv(), MatrixSave(), MatrixShow().  
Matrix Inverse: Compute the inverse of the N×N matrix. N is in {2,3,4}. The N² elements of the matrix are in the mtx argument, rowwise and separated by commas. See also MatrixDet(), MatrixSave(), MatrixShow().  
Matrix InvMod: Given the N×N input matrix (in mtx), compute the inverse (modulo mod) matrix. N is in {2,3,4}. The N² elements of the matrix are in the mtx argument, rowwise and separated by commas. Typically this mulitplicative inverse matrix is used with HillCipher(). MatrixSave() also computes the inverse modulo mod matrix, specifically for use with HillCipher(). See also MatrixDel(), MatrixShow().  
Matrix Multiply: Display the results of multiplying the two N×N matrices (mtx1 and mtx2). N is in {2,3,4}. The data for the two matrices are entered in rowwise order and separated by commas.  
Matrix Multiply (Ids): Multiply saved matrix (ID=mtxId1) and saved matrix (ID=mtxId2), display the results, and save them as matrix (ID=mtxId3). The mtxIds are in (1–20), and must all be different. See also MatrixSave() to create a matrix, and MatrixShow() to review the matrix data.  
Matrix Save: Save the N×N input matrix as id mtxId for later use. N is one of {2,3,4}. mtxId is in (120). When modId is zero, just save the matrix which may have real and integer values. Otherwise, modId is one of {1,2,3} for use with HillCipher() with corresponding alphabets of {31,47,73} characters (and the modulus of the cipher). The N² elements of the array are in mtx in row order, separated by commas. For modId not zero, the matrix entries must be integers in the range of zero to (alphabetSize1), and the inverse modular matrix is internally generated for use with HillCipher() decoding. See also MatrixShow() and especially HillCipher() for details of using the specialcase allinteger modId matrix.  
Matrix Show: Display the saved matrix with id mtxId. If mtxId is zero, display a list of the attributes of all the saved matrices. See also MatrixSave(), HillCipher().  
Matrix Transpose: Save and display Matrix (ID=mtxTr) as the transpose of Matrix (ID=mtxIn).The matrix IDs must be different, and mtxIn must exist. If mtxTr already exists, it is silently replaced. See also MatrixSave(), MatrixShow(), MatrixDel().  
Matrix Vector Multiply: Multiply the N×N matrix (ID=mtxIn) times the N×1 column vector (ID=vecIn) to produce the 1×N row vector (ID=vecOut). N is a property of the saved data in the range (24). See also VectorMtxMult(), MatrixMultId(), MatrixSave(), VectorSave().  
Mid Point: Given two data points on the XY plane, (x1, y1) and (x2, y2), display the coordinates of the midpoint of the line defined by the two points. See also Line(), Ymxb(), MidPt3D().  
Mid Point (3D): Given two data points in 3D space, (x1, y1, z1) and (x2, y2, z2), display the coordinates of the midpoint of the line defined by the two points. See also MidPt(), Line3D()  
Mileage: Keep track of gas mileage and cost – enter data at each fillup and get running totals and averages (e.g., miles/gal, miles/$, $/fillup, ...). VehID is the vehicleID (either 1 or 2), units is units (gal,mi, or l,km), dist is distance (mi or km) traveled since last fillup, amt is the amount (gal,l) to fill tank, and cost is the cost of the fillup. The dollar sign ($) is used as the universal currency symbol. When dist is zero, simply generate a report  
Mod: Mod(a,b) operates on two integers (a, b), and is defined as the amount by which a exceeds the largest integer multiple of b which is not greater than a. Thus, Mod(17,4) = 1, and Mod(17,4) = 3. Mod(a,b) is always ≥ 0. This version is extended to handle real numbers. Note that the CrunchJr implementation of the % operator follows the precedent set by Fortran (and thus C and C++) many years ago as the more familiar remainder from division.  
Momentum: Momentum is the product of a mass and its velocity. Momentum is a vector quantity, with both magnitude and direction (that of the mass' velocity). In a closed system, linear momentum is conserved (in both Newtonian and Relativistic settings). Momentum(m,v) = mv  
Mortgage: The Monthly Mortgage Payment for a loan of loan dollars at rate annual percent (e.g., 4.85, not 0.0485) for years years. Result is composed of principal and interest only; escrow (e.g., taxes, insurance) is not included in the computation. See also Refinance() and PayoffStrat() functions. Mortgage(P,R,Y) = (rP)/(1(1/(1+r)^{m})) where r = R/12, and m = Y×12  
 
Nominal Rate: The nominal rate for an effective annual interest rate (rate), compounded N times per year. (rate > 0), and N is an integer > 1. The 'nominal rate' is the rate per compound period. See also EffectiveRate().  
Normalize: Normalize a multielement input vector; length in (216). At least one element must be nonzero. Divide each element by the Sqrt(sum of the squares of the elements). See also VectorNorm()  
Normalize2: Normalize a multielement input vector; length in (216). At least one element must be nonzero. Divide each element by the largest of the absolute values of the elements  
Normalize3: Normalize a multielement input vector; length in (216). At least one element must be nonzero. Divide each element by sum of the absolute values of the elements  
 
Octal: Display the integer argument N in octal.  
Ohm's Law: Compute electrical parameters. Set two of the four arguments to zero and two to nonzero to compute the value of the two zero arguments: P = Power (watts) V = Voltage (volts) I = Current (amps) R = Resistance (ohms)  
 
Payoff Strategy: Investigate various loan characteristics and payoff strategies for New or Existing mortgages. This MiniApp works with basic payment data only (i.e., Principal and Interest, but not Escrow = Taxes+Insurance). In particular, one can review the impact of paying an extra amount with each monthly payment, the impact of paying a onetime lump sum payment, and the impact of paying ½ of the regular monthly payment every other week. The View and Profile strategies show payoff characteritics of a loan, and the Survey strategy displays a variety of extra and lump sum impacts. The Lump sum and Profile options are for Existing loans only.  
Pendulum Period: Period (seconds) of a pendulum of length len (feet). PendPeriod(x) = 2π√ x/g  
Permutations: The number of different ways that m things may be selected from n things. Both arguments must be integers. and (m < n) Permutations(n,m) = n!/(nm)!  
Plane2: Given two planes in the form Ax + By + Cz + D = 0, display the angle between them and the equation of the line at their intersection. Same as IntersectPP(). See also the other Plane*() and the Intersect*() family of functions  
Plane Equations: Show various renditions of the equation of the plane: "Ax + By + Cz + D = 0." At least 1 of {A,B,C} must be nonzero.  
Plane Line: Show the intersection coordinates of the line defined by ( x1, y1, z1 ) and ( x2, y2, z2 ) and the plane defined by Ax + By + Cz + D = 0. Same as IntersectLP(). See also the other Plane*() and the Intersect*() family of functions.  
Plane Point: Given a point ( x, y, z ) and a plane defined by 'Ax + By + Cz + D = 0', show the distance between them, the intersection point, and the normal vector from the plane to the point. See also DistPtPlane() and the other Plane*() and Intersect*() family of functions  
Plane Point 2: Given 2 points ( x1, y1, z1 ) and ( x2, y2, z2 ) show the equation of the plane which passes through point 1 and perpendicular to the line formed by the 2 points. See also the other Plane*() and the Intersect*() family of functions  
Plane Point 3: Given 3 points, show the plane they define and its normal vector. See also the other Plane*() and Intersect*() family of functions  
Polar: The polar coordinates (distance from origin and angle (radians)) of the point defined by the cartesian coordinates (x, y)  
Potential Energy: Potential Energy of an object of mass mass at rest at a height of height. For Imperial units, PEnergy, mass, height are {ftlbf, lb, ft}; for Metric, {J, kg, m}. PEnergy(m,h) = gmh where g is the appropriate standard gravity  
Power: Power is the rate of doing work. Power(W,t) = W/t  
Present Value: The present quantity which will accumulate to the Future Value value in yrs years at an annual interest rate of rate percent compounded N times per year. If N is zero, simple interest is used. See also FutureValue().  
Price Per: Display the price per unit ratios p1/n1, p2/n2, ... (2 to 6 pairs) for comparison purposes. n1, n2, ... ≠ 0. Similar to Ratios(). Sample input: 19.79,12, 99.95,75, 4.49,2  
Product: The product of the integers (1,2,...,N). Same as Factorial() except: Product(0) = 0, Factorial(0) = 0! = 1  
Pythagorean: Length of the Hypotenuse of a right triangle with sides a and b Pythagorean(a,b) = +√ a² + b²  
 
Quadratic Equation: The roots to the 2^{nd} order equation: ax² + bx + c = 0 (a ≠ 0)  
Quadrature: _{a}∫^{ b}{expr(x)} dx. Approximate result of integrating the expression (expr) from a to b. A and b are the lower and upper bounds of the (dummy) variable in expr, and must be constants with (a ≠ b). The expression expr must have exactly one variable, must be singlevalued, and may be a single userdefined function or an expression with no userdefined functions. The expression expr must be continuous over the specified range, and may include any singlevalued function except {Rand(), IRand(), CoinFlip(), RockPS(), RockSpock(), Dice()}. Unpredictable results may occur when expr is not continuous or is otherwise illbehaved. Same as Integrate(), except Quadrature() always shows convergence of the approximate solution. Quadrature() is strongly recommended.  
Quartic Equation: The roots to the 4^{th} order equation: ax^{4} + bx^{3} + cx^{2}+ dx + e = 0 (a ≠ 0)  
 
Race Pace: Pace (mm:ss per mi/km) for race of distance dist miles/kilometers run in hr hours, min minutes, and sec seconds. Distance is in miles for Imperial, and kilometers for Metric. Note that dist may be a constant or an expression, e.g., MARATHON, TENK, FIVEK, 100/1760, 0.8*KM, METRICMILE, HALFMARATHON, etc  
Race Time: Time (hh:mm:ss) to run a race of distance dist miles/kilometers run at a pace of min minutes and sec seconds per mile/kilometer. Distance is in miles for Imperial, and kilometers for Metric. Note thatdist may be a constant or an expression, e.g., MARATHON, TENK, FIVEK, METRICMILE, etc  
Radians: Convert d (degrees) to radians. See also Degrees() Radians(d) = πd/180 = d × D2RADIANS ≈ d × 0.01745329  
Rand: Return a random floating point value between x and y. Note  not reproducible. See also IRand(), CoinFlip(), RockPS(), Dice().  
Rankine: Convert Fahrenheit temperature (x) to Rankine. See also Kelvin(). Rankine(x) = x + 459.67  
Rates: Display the average time for an action when numb actions take hrs hours, mins minutes, and secs seconds.  
Rates2: Display the average time for an action when numb actions take yrs years, days days, and hrs hours.  
Ratios: Display the ratios v1/n1, v2/n2, ... (2 to 6 pairs) for comparison purposes. n1, n2, ... ≠ 0. Similar to PricePer().  
Refinance: Impact (total cost, breakeven, % interest, et cetera) of refinancing a loan: • currBalance = Current loan balance (principal) • currPayment = Current loan payment (principal and interest components only) • currRate = Current loan annual interest rate (percent  e.g., 7.125) • newRate = New loan annual interest rate (percent  e.g., 5.375) • loanCharge = Addition to principal (loan charge  e.g., closing costs, points) • loanLoan = Addition to principal ("equity" payout  may be 0) • newDuration = New loan duration (years) Notes: (1) Do not include escrow payments (typically taxes and insurance) with currPayment, (2) See also Mortgage() and PayoffStrat() functions  
Regular Polygon: Information about the nSidessided regular polygon with side length. A regular polygon has all equal angles  which implies all equal sides (Note: all equal sides does not imply equal angles). Name, side, area, perimeter, and the radii of the inscribed and circumscribed circles are given. See also RegPolygon2()  
Regular Polygon2: Information about the nSidessided regular polygon with radius radius. Name, angle, area, and the sideLength of the inscribed and circumscribed polygon are given. See also RegPolygon()  
Resistance: Resistance for 210 resistors (r1, r2, ...) in serial, parallel. All args > 0.  
RGB Display: Display the Red, Green, and Blue components of color in a table. Also, display the HSV representation and an iPhone/iPodTouch/iPad SDK ObjectiveC format representation. Color is an integer representation of the color (e.g., 0x66F739), and is silently kept in the range (0  0xFFFFFF).  
RockPaperScissors: Simulate a random RockPaperScissors (123) selection.  
RockSpock: Simulate a random RockPaperScissorsLizardSpock (12345) selection. This game is an extended form of RockPaperScissors (suggested by Sheldon Cooper (of CBSTV's The Big Bang Theory)). Outcomes: Rock defeats (Lizard,Scissors); Paper d. (Rock,Spock); Scissors d. (Paper,Lizard); Lizard d. (Spock,Paper). Spock d. (Scissors,Rock).  
Roots: The n roots of x. n is an integer in (2‐12)  
 
Sin: Sine of the angle x  
Sinh: Hyperbolic sine of the angle x Sinh(x) = ½(e^{x} – e^{–x})  
Solver: Solve N equations in N unknowns (2 ≤ N ≤ 4). There are (N*(N+1)) more arguments, representing the equations. Solver() is the same as Unknowns(). For example: x + 2y + 3z = 4 2x + y + 4z = 5 3x + 3y + 6z = 12 For these equations, using Solver(3, 1,2,3,4, 2,1,4,5, 3,3,6,12) yields: x = 7, y = 3, z = 3  
Sphere4: Given 4 points in 3D space, determine the center and the radius of the sphere they define. The arguments are the 4 points: x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4.  
Square Root (Sqrt): The positive square root of x, (x ≥ 0) Sqrt(x) = +√ x  
Statistics: Report statistics about the arguments. Must have 3 or more arguments. Show: Number, Average, Standard Deviation, Median, Sum, Sum of the Squares, Max and MaxLoc, Min and MinLoc  
Sum: Sum of two or more values  
Sum Cubed: Sum of the cubes of the two or more arguments  
Sum Inverses: Sum of the inverses of the two or more values. All values must be nonzero.  
Sum Inverses Cubed: Sum of the inverses cubed of the two or more values. All values must be nonzero.  
Sum Inverses Squared: Sum of the inverses squared of the two or more values. All values must be nonzero.  
SumKxK: ∑_{k=1,n}{k(x^{k})} (x ≠ 0, 1)  
Summation: The sum of the integers (1,2,...,N).  
SumNCubed: The sum of the cubes of the integers (1,2,...,N).  
SumNSquared: The sum of the squares of the integers (1,2,...,N).  
SumofPowers: ∑_{m=1,n}{m^{p}} where n and p are both integers, (n > 1), and (0 < p < 11)  
Sum Squared: Sum of the squares of the two or more arguments  
 
Tan: Tangent of the angle x  
Tangent Circle: Given the radii of 3 mutually tangent circles (r1, r2, r3), return the radius of a 4^{th} circle which is tangent to all 3 input circles  
Tanh: Hyperbolic tangent of the angle x. Tanh(x) = (e^{x} – e^{–x})/(e^{x} + e^{–x})  
Three Doors: Simulate a simple but nonintuitive game show. Tap Explain for an explanation of how he game works, or tap Play and pick a Door to start the game.  
Tip: Array of tip candidates for bill bill. See also BillSplit().  
Trig 101: A collection of 24 commonlyoccurring sin(), cos(), and tan() functions. Enter one argument and get all 24 results, as well as some basic trig facts.  
Trig 102: A collection of 12 regular, inverse, and hyperbolic trigonometric functions. Enter one argument and get all 12 results, as well as some additional trig facts.  
 
Unknowns: Solve N equations in N unknowns (2 ≤ N ≤ 4). There are (N*(N+1)) more arguments, representing the equations. Unknowns() is the same as Solver(). For example: w + x + y + z = 10 w + 2x + y + 6z = 32 2w + y + z = 9 3w + 2x + 5y + 4z = 38 For these equations, using Unknowns(4, 1,1,1,1,10, 1,2,1,6,32, 2,0,1,1,9, 3,2,5,4,38) yields: w = 1 y = 3 x = 2 z = 4  
 
Vector Copy: Copy the data for saved vector (Id=inId) to vector (Id=newId). (newId ≠ inId). The id's are in (120). See also VectorDel(), VectorSave(), VectorShow().  
Vector Delete: Delete the saved vector with id = id. See also VectorSave(), VectorShow(), VectorCopy().  
Vector Dot Product: Compute the vector dot product of the two saved vectors (id1, id2). The vectors must be the same length. See also VectorSave(), VectorShow().  
Vector Matrix Multiply: Multiply the N×1 row vector (ID=vecIn) times the N×N matrix (ID=mtxIn) to produce the 1×N column vector (ID=vecOut). N is a property of the saved data in the range (24). See also MatrixVecMult(), MatrixMultId(), MatrixSave(), VectorSave().  
Vector Normalize: Create vector (Id=nmId) as a normalized copy of vector (Id=inId). At least one element must be nonzero. Divide each element by the Sqrt(sum of the squares of the elements). The id's need not be different.  
Vector Save: Save the N×1 input vector as Id = id for later use. N is in (2–16). id is in (1–20). The final argument contains the N vector elements. After saving the vector, display the attributes of its elements. See also VectorDel(), VectorShow(), VectorCopy().  
Vector Show: Display the saved vector with Id = id. If id is nonzero, display the attributes of the elements of vector id. If id is zero, display an abbreviated list of the attributes of all the saved vectors. See also VectorSave(), VectorDel(), VectorCopy().  
Vector Sort: Save Vector (ID=sortId) as a sorted copy of Vector (ID=inId), and display its contents. The id's must be in (1–20) and need not be different. Sorting may be ascending or descending, and duplicates may be kept or discarded.  
Vector Transpose: Save Vector (ID=trId) as the transpose of Vector (ID=inId), and display its contents. The id's must be in (1–20) and need not be different.  
Volume Brick: Volume of the rectangular solid with the sides length, width, and depth. VolBrick(L,W,D) = LWD  
Volume Cone: Volume of a right cone of radius radius and height height. VolCone(r,h) = πr²h/3  
Volume Cube: Volume of a cube of side s > 0. VolCube(s) = s³  
Volume Cylinder: Volume of a right cylinder of radius r and height h (both > 0). VolCylinder(r,h) = πr²h  
Volume Frustum: Volume of the curved surface of the frustum of a right cone with base radius baseRadius, top radius topRadius, and height height (all > 0). VolFrustum(b,t,h) = πh(b²+bt+t²)/3  
Volume Oblate Spheroid: Given an ellipse with major semiaxis major and minor semiaxis minor, (both > 0; major ≥ minor), an oblate spheroid is formed by the rotation of the ellipse about its minor axis. The volume of an oblate spheroid is given by: VolOblSpheroid(a,b) = (4/3)πa²b  
Volume Parallelepiped: Volume of a parallelepiped of sides a, b, c and internal angles between them r, s, t (degrees) (all data > 0). VolPpiped(a,b,c, r,s,t) = abc × Sqrt{ 1 + 2Cos(r)Cos(s)Cos(t) – Cos²(r) – Cos²(s) – Cos²(t) }  
Volume Prolate Spheroid: Given an ellipse with major semiaxis major and minor semiaxis minor, (both > 0; major ≥ minor), a prolate spheroid is formed by the rotation of the ellipse about its major axis. The volume of an prolate spheroid is given by: VolProlSpheroid(a,b) = (4/3)πab²  
Volume Pyramid: Volume of a regular pyramid of base area base and height ht (both > 0). VolPyramid(b,a) = ba/3  
Volume Spherical Cap: Volume of a spherical cap of sphere radius r and height h (both > 0; h < 2×r). SphR is the radius of the sphere, not the cap. A spherical cap is also known as a dome. See also VolSphCap2(). VolSphCap(r,h) = πh²(3rh)/3  
Volume Spherical Cap2: Volume of a spherical cap of cap radius r and height h (both > 0). r is the radius of the cap, not the sphere. A spherical cap is also known as a dome. See also VolSphCap(). VolSphCap2(b,h) = πh(h²+3b²)/6  
Volume Sphere: The volume of a sphere of radius r > 0. VolSphere(r) = (4/3)πr³  
 
WindChill: Display the windchill factor (°F) for a given temperature temp and wind speed speed.The Windchill factor is the apparent temperature on exposed skin due to the wind.  
Work: Work is the product of a force and the distance through which it acts (only the distance traveled in the direction of the force). W(F,d) = Fd  
 
Y = mx+b: Given two data points on the XY plane, (x1, y1) and (x2, y2), compute the slope (m), the yintercept (b), the distance between the two points, the midpoint, and show the equation of the line containing the points. Same as Line(). See also Line3D().  
