# Chapter 4. Expressions

Expressions are mathematical statements. Mathematical questions are asked through expressions, which contains objects tied together with operators. The result of an expression may also be an expression, if the result is not a single object. Apples and oranges can be mixed, but the result will hold them apart. Qalculate! knows algebra.

## Objects

In Qalculate! mathematical entities, such as numbers and variables, are referred to as objects. The recognized object types are listed below.

Numbers

These are the regular numbers composed by digits 0-9 and a decimal sign — a dot, or a comma if it is the default decimal point in the locale/language used. If comma is used as decimal sign, the dot is still kept as an alternative decimal sign, if not explicitly turned off in the preferences dialog with (to allow it to be used as thousand separator instead). Numbers include integers, real numbers, and complex numbers. The imaginary part of complex numbers are written with as regular number followed by the special variable i (can be changed to a j, placed in front of the imaginary part, in the preferences dialog), which represents the square root of -1 (e.g. 2 + 3i). Spaces between digits are ignored (5 5 = 55). E (or e) can be considered as a shortcut for writing many zeroes and is equivalent to multiplication by 10 raised to the power of the right-hand value (e.g. 5E3 = 5000).

Sexagesimal numbers (and time) can be entered directly using colons (e.g. 5:30 = 5.5). A number immediately preceded 0b, 0o, 0d or 0x are interpreted as a number with base 2, 8, 12 or 16, respectively (if the default base is 10, e.g. 0x3f = 63). The number base in can also be selected, either by using the base(), bin(), oct(), hex() and roman() functions, or by setting the base used for all numbers in the whole expression from ModeNumber BaseSelect Result and Expression Base.... For details about supported number bases see Table 5.1, “Supported Number Bases”.

Intervals

A number interval can be entered using the interval() function (specifies the upper and lower limit of the interval), the uncertainty() function (specifies relative or absolute uncertainty), or using ± or +/-, specifying the width of the interval after the mid value (e.g. 5±1 = uncertainty(5, 0.2) = interval(4, 6)). If the read precision option is activated, decimal numbers are interpreted as an interval between the numbers that are normally rounded to the entered number (e.g. 1.1 = 1.1±0.05). If interval calculation using variance formula is activated (default), the interval represents the standard uncertainty (deviation) of the value.

Vectors and Matrices

A matrix is a two-dimensional rectangular array of mathematical objects. Vectors are matrices with only one row or column, and thus one-dimensional sequences of objects. Vectors and matrices are generated by vector(), matrix() and similar functions, or using a syntax in the form of [1 2 3 4] and [1 2; 3 4], with columns separated by space or comma and rows separated by semi-colon, or (1, 2, 3, 4) and ((1, 2), (3, 4)). Vectors are generally considered as matrices with one row (row vector) in operations that expect a matrix (e.g. matrix multiplication). Matrices and vectors with many elements are easier to handle if stored in variables.

Variables/Constants
Functions
Units and Prefixes

Qalculate! understands abbreviated, plural and singular forms of unit names and prefixes. Prefixes must be put immediately before the unit to be interpreted as prefixes — 5 mm = 0.005 m, but 5 m m = 5 m^2. Also, for convenience units allow the power operator to be left out. A number following immediately after a unit is interpreted as an exponent (e.g. 5 m2 = 5 m^2). This does not apply to currencies, as they might be put in front of the quantity. More information in Chapter 9, Units.

Unknowns

Unknowns are text strings without any associated value. These are temporary unknown variables with default assumptions. Unknowns can also be explicitly entered by placing a backslash (\) before a single character (e.g. 5\a + 2\b) or using quotation marks (" or ') before and after a text string (e.g. 5 "apples" + 2 "bananas"). If unknowns are activated (Mode+Enabled ObjectsUnknowns) and Qalculate! finds a character that are not associated with any variable, function or unit in an expression, then it will be regarded as an unknown variable. See Chapter 7, Variables.

Date and Time

Date/time values are specified using quoted text string (quotation marks are not needed for function arguments), using standard date and time format (YYYY-MM-DDTHH:MM:SS). Some local formats are also supported, but not recommended. The local time zone is used, unless a time zone is specified at the end of the time string (Z/UTC/GMT or +/-HH:MM). Date/time supports a small subset of arithmetic operations. The time units represents calendar time, instead of average values, when added or subtracted to a date.

Text

This category represent a number of different function argument types, such as regular text and file names. They can, but do not need to be put in quotes except when containing the argument separator (, or ;).

All text after a hashtag (e.g. (5×2)/2 #calculating triangle area) is treated as a comment, which are added to the history. Use double hashtags (##) at the beginning of the expression to add a comment as a separate history item at the top.

To avoid confusion, functions, units, variables and unknown variables can independently be disabled.

Variables, functions and units are all accessible in the menus and in the variable, function and unit managers, If their names are not remembered. Functions accessed this way has some extra conveniences. If the function has at least one argument, a dialog will pop up where arguments can be entered and a description of the function and its arguments is available.

Qalculate! can handle most commonly used symbols for certain variables, functions and units, even though most are difficult to find on a keyboard. These include π for pi, √ for sqrt, € for euro, and so on. Most importantly it is possible to copy these symbols when used in the result.

For more information about variables, functions and units, see Chapter 7, Variables, Chapter 8, Functions and Chapter 9, Units.

## Operators

The following operators are defined in Qalculate! and may be used in expressions. Word operators (such as AND) must be surrounded by space (e.g. 5 mod 2, not 5mod2.

Table 4.1. Operators

Operation

Symbol

Description

Example

Result

+, plus

Adds the right value to the left value.

1 + 1

2

Subtraction

−, minus

Subtracts the right value from the left value.

1 − 1

0

Multiplication

×, ⋅, *, times

Multiplies the left value by the right value.

2 × 2

4

Division

/, per

Divides the left value by the right value.

2 / 2

1

Remainder

%, rem

Returns the remainder after division.

3%2

1

Modulo

%%, mod

Returns the remainder after division.

3 mod -2

-1

Integer Division

//, \, div

Divides the left value by the right value and rounds the result towards zero.

5 // 2

2

Exponentiation

^, **

Raises the left value by the right value. Can also be typed as **. Note that x^y^z equals x^(y^z), and not (x^y)^z. Note also that for non-integer exponents with negative bases, the principal root is returned and not the real root ((-8)^(1/3) equals 1 + 1.73i instead of -2). To calculate the real root for negative values, use the cbrt() and root() functions.

2^3

8

10^x

E

Multiplies the left value with 10 raised to the power of the right value. Equivalent to the exponential number format in result display. E is as much an operator as part of numbers.

1E3

1000

Factorial

!

Returns the factorial of the value to the left of the operator. If the operator is repeated the corresponding multifactorial is returned.

5!

120

Parenthesis

( and )

Evaluates the expression in parenthesis first.

5 × (1 + 1)

10

Parallel sum

∥, ||

Returns the the reciprocal value of a sum of reciprocal values. || is interpreted as parallel if units are used, otherwise as logical OR.

10 Ω || 6 Ω

3.75 Ω

Equals

=

Returns true if the left value equals the right value. Unknown variables (e.g. x) are isolated if the expression does not evaluate as true or false.

1 = 2, 5x = 5

1, x=1

Not equals

≠, !=

Returns true if the left value does not equals the right value. Unknown variables (e.g. x) are isolated if the expression does not evaluate as true or false.

1 != 2, x + 2 != 5

1, x != 3

Less than

<

Returns true if the left value is is less than the right value. Unknown variables (e.g. x) are isolated if the expression does not evaluate as true or false.

1 < 2

1

Greater than

>

Returns true, if the left value is greater than the right value. Unknown variables (e.g. x) are isolated if the expression does not evaluate as true or false.

1 >2

0

Less than or equal

≤, <=

Returns true if the left value is less than or equal the right value. Unknown variables (e.g. x) are isolated if the expression does not evaluate as true or false.

1 <= 2

1

Greater than or equal

≥, >=

Returns true if the left value is greater than or equal the right value. Unknown variables (e.g. x) are isolated if the expression does not evaluate as true or false.

1 ≥ 2, x + 5 ≥ 7

0, x ≥ 2

Logical NOT

!, not

Returns true if the value to the right is false.

!(1>2)

1

Logical OR

||, or

Returns true if the right or left value is true.

1>2 || 2>1

true

Logical XOR

⊕, xor

Returns for true if one, but not both, of the right or left value is true.

1>2 ⊕ 2>1

true

Logical NOR

nor

Returns true if both the right and left value is false.

1>2 nor 2>1

false

Logical AND

&&, and

Returns true if both the right and left value is true.

1>2 && 2>1

false

Logical NAND

nand

Returns true if the right or left value is false.

1>2 nand 2>1

true

Bitwise NOT

¬, ~

Equivalent to -1 − x.

~(0010 | 1100)

-1111

Bitwise Shift Left

<<

Shifts the bits of the left value x steps to the left, where x is the value on the right. Implemented as a shortcut for shift()

0011 << 1

0110

Bitwise Shift Right

>>

Shifts the bits of the left value x steps to the right, where x is the value on the right. Implemented as a shortcut for shift()

0011 << 1

0001

Bitwise OR

∨, |

If a bit is 1 in one of the numbers set it to 1, otherwise 0. Also functions as elementwise logical operator on vectors.

0010 | 1100

1110

Bitwise XOR

⊻, ^^, xor

If a bit is 1 in one of the numbers and not in the other, set it to 1, otherwise 0. Can normally also be used as logical XOR. ⊻ can be input using Ctrl+^ (or just ^ if selected in preferences) on the keyboard.

1010 ⊻ 1100

0110

Bitwise AND

∧, &

If a bit is 1 in both numbers set it to 1, otherwise 0. Also functions as elementwise logical operator on vectors.

1010 & 0011

0010

Dot Product

., dot

Returns the dot product for two vectors.

[1, 2, 3].[4, 5, 6]

32

Cross Product

⨯, cross

Returns the cross product for two vectors.

[1, 2, 3] cross [4, 5, 6]

[-1, 6, -3]

Elementwise Multiplication

.×, .*

Multiplies each element of a vector/matrix with the corresponding element in another vector/matrix, or a scalar.

[1, 2, 3].*[4, 5, 6]

[4, 10, 18]

Elementwise Division

./

Divides each element of a vector/matrix by the corresponding element in another vector/matrix, or a scalar.

[2, 4, 6]./2

[1, 2, 3]

Elementwise Exponentiation

.^

Multiplies each element of a vector/matrix with the corresponding element in another vector/matrix, or a scalar.

[1, 2, 3].^2

[1, 4, 9]

Transpose

.'

Returns the transpose of the matrix to the left of the operator.

[[1, 2, 3], [3, 4, 5]].'

[[1, 3], [2, 4], [3, 5]]

Combination

comb

Same as comb() function.

5 comb 2

10

Permutations

perm

Same as perm() function.

5 perm 2

20

Save as Variable/Function

:=, =

Saves the value or expression to the right of the operator as a variable or function (as save() function). If colon is omitted, the expression is calculated before it is assigned to the variable.

var1:=5

func1():=x+y

var1=ln(5)+2

The multiplication sign can generally be left out. This is not true for numbers (5(5) = 25 but 5 5 = 55). Expressions can also generally be written with or without spaces with the same result (2xsin(2) equals 2 x sin(2) which equals 2 × x × sin(2)), but be careful. The vast number of functions and units means that without separating spaces, the result might not be obvious. To avoid confusion Qalculate! can limit the use of implicit multiplication (ModeLimit Implicit Multiplication), so that space, operator or parenthesis must be put between functions, units and variables (in this mode esqrt(5) does not equal e × sqrt(5)). Also note that unit prefixes must be put immediately before the unit, to be interpreted as prefixes (5 mm = 0.005 m, but 5 m m = 5m^2). You can see how to expression was interpreted in the history window.

Usually, mathematical expressions are written as normally expected. Standard operator precedence apply. Expressions are evaluated according to the following priorities:

1. Parenthesis

2. E (10^x)

3. Exponentiation (x^y)

4. Functions (e.g. sqrt(2))

5. Bitwise NOT (~)

6. Logical NOT (!)

7. Element-wise operations, and cross and dot products (.*, ./, .^, ., ⨯)

8. Multiplication, division, integer divison, remainder, modulo (*, /, //, %, %%)

9. Parallel sum (∥)

10. Addition and subtraction (+, −)

11. Bitwise NOT (~)

12. << and >>

13. Comparison (>, <, =, >=, <=)

14. Bitwise AND (&)

15. Bitwise XOR (⊻)

16. Bitwise OR (|)

17. Logical AND (&&)

18. Logical NAND

19. Logical NOR

20. Logical OR (||)

21. Logical XOR (⊕)

## Implicit Multiplication and Parsing Modes

The evaluation of short/implicit multiplication, without any multiplication sign (e.g. 5x, 5(2+3)), differs depending on the parsing mode. In the conventional mode implicit multiplication does not differ from explicit multiplication (12/2(1+2) = 12/2×3 = 18, 5x/5y = 5 × x/5 × y = xy). In the parse implicit multiplication first mode, implicit multiplication is parsed before explicit multiplication (12/2(1+2) = 12/(2 × 3) = 2, 5x/5y = (5 × x)/(5 × y) = x/y). The default adaptive mode works as the parse implicit multiplication first mode, unless spaces are found (1/5x = 1/(5 × x), but 1/5 x = (1/5) × x). In the adaptive mode unit expressions are parsed separately (5 m/5 m/s = (5 × m)/(5 × (m/s)) = 1 s). Function arguments without parentheses are an exception, where implicit multiplication in front of variables and units is parsed first regardless of mode (sqrt 2x = sqrt(2x)).

If the limit implicit multiplication option is activated, the use of implicit multiplication when parsing expressions and displaying results will be limited to avoid confusion. For example, if this mode is not activated and integrte(5x) is accidently typed instead of integrate(5x), the expression is interpreted as int(e × e × (5 × x) × gr × t) (displayed in history window). The result will then without any error be int(2.3940139x × km^2) instead of 2.5x^2. If limit implicit multiplication is activated, the mistyped expression would instead show an error telling that integrte is not a valid variable, function or unit (unless unknowns is enabled in which case the result will be 5 "integrate" × x). When implicit multiplication is limited, variables, functions and units must be separated by a space, operator or parenthesis (xy does not equal x × y).

In addition there are two special parsing modes — RPN syntax (for details see the section called “The RPN Mode”) and chain syntax. The chain syntax interprets expressions in a manner similar to the immediate execution mode of a traditional calculator. Instead of using the standard order of operations, the expression is simply calculated from left to right (e.g. 1 + 2 × 3 = (1 + 2) × 3 = 9 instead of 1 + 2 × 3 = 1 + (2 × 3) = 7). Functions, with a simgle argument, applies to the value immediate to the left of the function name (e.g. 1 + 2 sin = 1 + sin(2)), unless parentheses are used.

## The “to” (and “where”) operators

Putting to (or a right arrow, e.g. ->) followed by an expression at the end of the mathematical expression is mainly used for unit conversion (see the section called “Conversion”). There are however also some convenient commands that can be typed after to . Here is a list of possible to values:

A unit or unit expression

Convert to a unit or a unit expression (e.g. 5 ft + 2 in to meter = 1.5748 m or 50 mph to km/h ≈ 80 km/h). Prepend with a question mark (?) to request the optimal prefix. Modifiers in front of the question mark selects the type of prefixes used — 'b' for binary prefixes, 'd' for decimal prefixes, and 'a' for all decimal prefixes incluing centi, deci, etc. (e.g. 8 × 1024 bits to b?bytes = 1 kibibyte). Prepend with + or - to force/disable use of mixed units (e.g. 5 m to + ft ≈ 5 yd + 1 ft + 4.9 in).

A physical constant or a variable

Convert to variable as unit (e.g. 500 km/ms to c ≈ 1.7 c).

base

Convert to base units (e.g. 1 lux to base = 1 cd/m2).

optimal

Convert to optimal unit (e.g. (10 J)/(2 s) to optimal = 5 W).

mixed

Convert to mixed units (e.g. 90 s to mixed = 1 min + 30 s.

bin / binary

Show as binary number (e.g. sqrt(900) to bin = 0001 1110). Append an integer to specify the number of bits (e.g. 4 to bin16 = 0000 0000 0000 0100).

oct / octal

Show as octal number (e.g. 52 to octal = 64).

duo / duodecimal

Show as duodecimal number (e.g. 152 to duo = 108).

Show as hexadecimal number (e.g. 623 to hex = 026F). Append an integer to specify the number of bits (e.g. 4 to hex16 = 0004).

sexa / sexa2 / sexa3 / sexagesimal

Show as sexagesimal number (e.g. 7.33 to sexagesimal = 7°19′48″). For sexa2, arcseconds are hidden, and for sexa3 arcseconds are rounded.

longitude / longitude2 / latitude / latitude2

Show as sexagesimal latitude/longitude (e.g. -7.33 to latitude = 7°19′48″S). longitude2/latitude2 only shows degrees and arcminutes (e.g. -7.33 to latitude2 = 7°19.8′S).

bijective

Show as bijective base-26 number (e.g. 731 to bijective = ABC).

binary16 / fp16, binary32 / float / fp32, binary64 / double / fp64, fp80, binary128 / fp128

Show as binary representation of IEEE 754 16-bit (half precision), 32-bit (single precision), 64-bit (double precision), 80-bit (x86 extended format), or 128-bit (quadruple precision) floating-point number.

time

Show in time format (e.g. 7.25 to time = 7:15.

roman

Show as Roman numerals (e.g. 1984 to roman = MCMLXXXIV).

Unicode

Show as unicode character(s) (uses UTF-32 for conversion, e.g. 0x178 to unicode = Ÿ).

base #

Show using the specified base (e.g. 523 to base 20 = 163 or circumference(1) to base pi = 20).

bases

Show as binary, octal, decimal, duodecimal, hexadecimal and Roman number (opens convert bases dialog with the mathematical expression).

rectangular / cartesian

Show complex number in rectangular form (e.g. 0.28i − 2 to complex = 0.28i − 2).

exponential

Show complex number in exponential form (e.g. 0.28i − 2 to exponential ≈ 2e^(3i)).

polar

Show complex number in polar form (e.g. 0.28i − 2 to polar ≈ 2(cos(3) + i × sin(3))).

angle / phasor

Show complex number in angle/phasor notation (e.g. 0.28i − 2 to angle ≈ 2∠3).

cis

Show complex number in cis form (e.g. 0.28i − 2 to angle ≈ 2 cis 3).

fraction

Show as mixed fraction 1.25 to fraction = 1 + 1/4.

partial fraction

Show expanded partial fractions (e.g. 1 / (x2 + 2x − 3) to partial fraction = 1 ∕ (4x − 4) − 1 ∕ (4x + 12)).

factors

Show factorized (algebraic or integer factorization, e.g. 3 645 678 to factors = 857 × 709 × 3 × 2 or x2 + 4x + 4 to factors = (x + 2)2).

calendars

Show date in different calendars (opens calendar conversion dialog).

UTC

Show date and time using UTC time zone.

UTC+/-hh[:mm]

Show date and time using specified time zone (e.g. UTC+08).

If to is not preceded by an expression, the previous result will be converted.

Similarly where (or alternatively /.) can be used at the end (but before to), for variable assignments, function replacements, etc. (e.g. x+y where x=1 and y=2, x^2=4 where x>0, and sin(5) where sin()=cos()).