8.1 Basic Arithmetic

The + (calc-plus) command adds two numbers. The numbers may be any of the standard Calc data types. The resulting sum is pushed back onto the stack.

If both arguments of + are vectors or matrices (of matching dimensions), the result is a vector or matrix sum. If one argument is a vector and the other a scalar (i.e., a non-vector), the scalar is added to each of the elements of the vector to form a new vector. If the scalar is not a number, the operation is left in symbolic form: Suppose you added ‘x’ to the vector ‘[1,2]’. You may want the result ‘[1+x,2+x]’, or you may plan to substitute a 2-vector for ‘x’ in the future. Since the Calculator can't tell which interpretation you want, it makes the safest assumption. See Reducing and Mapping, for a way to add ‘x’ to every element of a vector.

If either argument of + is a complex number, the result will in general be complex. If one argument is in rectangular form and the other polar, the current Polar mode determines the form of the result. If Symbolic mode is enabled, the sum may be left as a formula if the necessary conversions for polar addition are non-trivial.

If both arguments of + are HMS forms, the forms are added according to the usual conventions of hours-minutes-seconds notation. If one argument is an HMS form and the other is a number, that number is converted from degrees or radians (depending on the current Angular mode) to HMS format and then the two HMS forms are added.

If one argument of + is a date form, the other can be either a real number, which advances the date by a certain number of days, or an HMS form, which advances the date by a certain amount of time. Subtracting two date forms yields the number of days between them. Adding two date forms is meaningless, but Calc interprets it as the subtraction of one date form and the negative of the other. (The negative of a date form can be understood by remembering that dates are stored as the number of days before or after Jan 1, 1 AD.)

If both arguments of + are error forms, the result is an error form with an appropriately computed standard deviation. If one argument is an error form and the other is a number, the number is taken to have zero error. Error forms may have symbolic formulas as their mean and/or error parts; adding these will produce a symbolic error form result. However, adding an error form to a plain symbolic formula (as in ‘(a +/- b) + c’) will not work, for the same reasons just mentioned for vectors. Instead you must write ‘(a +/- b) + (c +/- 0)’.

If both arguments of + are modulo forms with equal values of ‘M’, or if one argument is a modulo form and the other a plain number, the result is a modulo form which represents the sum, modulo ‘M’, of the two values.

If both arguments of + are intervals, the result is an interval which describes all possible sums of the possible input values. If one argument is a plain number, it is treated as the interval ‘[x .. x]’.

If one argument of + is an infinity and the other is not, the result is that same infinity. If both arguments are infinite and in the same direction, the result is the same infinity, but if they are infinite in different directions the result is nan.

The - (calc-minus) command subtracts two values. The top number on the stack is subtracted from the one behind it, so that the computation 5 <RET> 2 - produces 3, not -3. All options available for + are available for - as well.

The * (calc-times) command multiplies two numbers. If one argument is a vector and the other a scalar, the scalar is multiplied by the elements of the vector to produce a new vector. If both arguments are vectors, the interpretation depends on the dimensions of the vectors: If both arguments are matrices, a matrix multiplication is done. If one argument is a matrix and the other a plain vector, the vector is interpreted as a row vector or column vector, whichever is dimensionally correct. If both arguments are plain vectors, the result is a single scalar number which is the dot product of the two vectors.

If one argument of * is an HMS form and the other a number, the HMS form is multiplied by that amount. It is an error to multiply two HMS forms together, or to attempt any multiplication involving date forms. Error forms, modulo forms, and intervals can be multiplied; see the comments for addition of those forms. When two error forms or intervals are multiplied they are considered to be statistically independent; thus, ‘[-2 .. 3] * [-2 .. 3]’ is ‘[-6 .. 9]’, whereas ‘[-2 .. 3] ^ 2’ is ‘[0 .. 9]’.

The / (calc-divide) command divides two numbers.

When combining multiplication and division in an algebraic formula, it is good style to use parentheses to distinguish between possible interpretations; the expression ‘a/b*c’ should be written ‘(a/b)*c’ or ‘a/(b*c)’, as appropriate. Without the parentheses, Calc will interpret ‘a/b*c’ as ‘a/(b*c)’, since in algebraic entry Calc gives division a lower precedence than multiplication. (This is not standard across all computer languages, and Calc may change the precedence depending on the language mode being used. See Language Modes.) This default ordering can be changed by setting the customizable variable calc-multiplication-has-precedence to nil (see Customizing Calc); this will give multiplication and division equal precedences. Note that Calc's default choice of precedence allows ‘a b / c d’ to be used as a shortcut for

     a b
     ---.
     c d

When dividing a scalar ‘B’ by a square matrix ‘A’, the computation performed is ‘B’ times the inverse of ‘A’. This also occurs if ‘B’ is itself a vector or matrix, in which case the effect is to solve the set of linear equations represented by ‘B’. If ‘B’ is a matrix with the same number of rows as ‘A’, or a plain vector (which is interpreted here as a column vector), then the equation ‘A X = B’ is solved for the vector or matrix ‘X’. Otherwise, if ‘B’ is a non-square matrix with the same number of columns as ‘A’, the equation ‘X A = B’ is solved. If you wish a vector ‘B’ to be interpreted as a row vector to be solved as ‘X A = B’, make it into a one-row matrix with C-u 1 v p first. To force a left-handed solution with a square matrix ‘B’, transpose ‘A’ and ‘B’ before dividing, then transpose the result.

HMS forms can be divided by real numbers or by other HMS forms. Error forms can be divided in any combination of ways. Modulo forms where both values and the modulo are integers can be divided to get an integer modulo form result. Intervals can be divided; dividing by an interval that encompasses zero or has zero as a limit will result in an infinite interval.

The ^ (calc-power) command raises a number to a power. If the power is an integer, an exact result is computed using repeated multiplications. For non-integer powers, Calc uses Newton's method or logarithms and exponentials. Square matrices can be raised to integer powers. If either argument is an error (or interval or modulo) form, the result is also an error (or interval or modulo) form.

If you press the I (inverse) key first, the I ^ command computes an Nth root: 125 <RET> 3 I ^ computes the number 5. (This is entirely equivalent to 125 <RET> 1:3 ^.)

The \ (calc-idiv) command divides two numbers on the stack to produce an integer result. It is equivalent to dividing with </>, then rounding down with F (calc-floor), only a bit more convenient and efficient. Also, since it is an all-integer operation when the arguments are integers, it avoids problems that / F would have with floating-point roundoff.

The % (calc-mod) command performs a “modulo” (or “remainder”) operation. Mathematically, ‘a%b = a - (a\b)*b’, and is defined for all real numbers ‘a’ and ‘b’ (except ‘b=0’). For positive ‘b’, the result will always be between 0 (inclusive) and ‘b’ (exclusive). Modulo does not work for HMS forms and error forms. If ‘a’ is a modulo form, its modulo is changed to ‘b’, which must be positive real number.

The : (calc-fdiv) [fdiv] command divides the two integers on the top of the stack to produce a fractional result. This is a convenient shorthand for enabling Fraction mode (with m f) temporarily and using ‘/’. Note that during numeric entry the : key is interpreted as a fraction separator, so to divide 8 by 6 you would have to type 8 <RET> 6 <RET> :. (Of course, in this case, it would be much easier simply to enter the fraction directly as 8:6 <RET>!)

The n (calc-change-sign) command negates the number on the top of the stack. It works on numbers, vectors and matrices, HMS forms, date forms, error forms, intervals, and modulo forms.

The A (calc-abs) [abs] command computes the absolute value of a number. The result of abs is always a nonnegative real number: With a complex argument, it computes the complex magnitude. With a vector or matrix argument, it computes the Frobenius norm, i.e., the square root of the sum of the squares of the absolute values of the elements. The absolute value of an error form is defined by replacing the mean part with its absolute value and leaving the error part the same. The absolute value of a modulo form is undefined. The absolute value of an interval is defined in the obvious way.

The f A (calc-abssqr) [abssqr] command computes the absolute value squared of a number, vector or matrix, or error form.

The f s (calc-sign) [sign] command returns 1 if its argument is positive, -1 if its argument is negative, or 0 if its argument is zero. In algebraic form, you can also write ‘sign(a,x)’ which evaluates to ‘x * sign(a)’, i.e., either ‘x’, ‘-x’, or zero depending on the sign of ‘a’.

The & (calc-inv) [inv] command computes the reciprocal of a number, i.e., ‘1 / x’. Operating on a square matrix, it computes the inverse of that matrix.

The Q (calc-sqrt) [sqrt] command computes the square root of a number. For a negative real argument, the result will be a complex number whose form is determined by the current Polar mode.

The f h (calc-hypot) [hypot] command computes the square root of the sum of the squares of two numbers. That is, ‘hypot(a,b)’ is the length of the hypotenuse of a right triangle with sides ‘a’ and ‘b’. If the arguments are complex numbers, their squared magnitudes are used.

The f Q (calc-isqrt) [isqrt] command computes the integer square root of an integer. This is the true square root of the number, rounded down to an integer. For example, ‘isqrt(10)’ produces 3. Note that, like \ [idiv], this uses exact integer arithmetic throughout to avoid roundoff problems. If the input is a floating-point number or other non-integer value, this is exactly the same as ‘floor(sqrt(x))’.

The f n (calc-min) [min] and f x (calc-max) [max] commands take the minimum or maximum of two real numbers, respectively. These commands also work on HMS forms, date forms, intervals, and infinities. (In algebraic expressions, these functions take any number of arguments and return the maximum or minimum among all the arguments.)

The f M (calc-mant-part) [mant] function extracts the “mantissa” part ‘m’ of its floating-point argument; f X (calc-xpon-part) [xpon] extracts the “exponent” part ‘e’. The original number is equal to ‘m * 10^e’, where ‘m’ is in the interval ‘[1.0 .. 10.0)’ except that ‘m=e=0’ if the original number is zero. For integers and fractions, mant returns the number unchanged and xpon returns zero. The v u (calc-unpack) command can also be used to “unpack” a floating-point number; this produces an integer mantissa and exponent, with the constraint that the mantissa is not a multiple of ten (again except for the ‘m=e=0’ case).

The f S (calc-scale-float) [scf] function scales a number by a given power of ten. Thus, ‘scf(mant(x), xpon(x)) = x’ for any real ‘x’. The second argument must be an integer, but the first may actually be any numeric value. For example, ‘scf(5,-2) = 0.05’ or ‘1:20’ depending on the current Fraction mode.

The f [ (calc-decrement) [decr] and f ] (calc-increment) [incr] functions decrease or increase a number by one unit. For integers, the effect is obvious. For floating-point numbers, the change is by one unit in the last place. For example, incrementing ‘12.3456’ when the current precision is 6 digits yields ‘12.3457’. If the current precision had been 8 digits, the result would have been ‘12.345601’. Incrementing ‘0.0’ produces ‘10^-p’, where ‘p’ is the current precision. These operations are defined only on integers and floats. With numeric prefix arguments, they change the number by ‘n’ units.

Note that incrementing followed by decrementing, or vice-versa, will almost but not quite always cancel out. Suppose the precision is 6 digits and the number ‘9.99999’ is on the stack. Incrementing will produce ‘10.0000’; decrementing will produce ‘9.9999’. One digit has been dropped. This is an unavoidable consequence of the way floating-point numbers work.

Incrementing a date/time form adjusts it by a certain number of seconds. Incrementing a pure date form adjusts it by a certain number of days.