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AddUp

A multi-purpose calculator

Basic Calculations

Basic arithmetic operations with AddUp are done by entering numbers and operators to form the expression to evaluate. The expression can be typed in and edited at will, then pressing the Enter key causes it to be evaluated. There is no need to use buttons with this calculator, everything that needs to be evaluated can be directly entered in the work area.

The basic arithmetic operators are:

• Addition: +
• Subtraction: -
• Multiplication: *
• Division: /

Other frequently-used operators are:

• Percent: % (for example, 5% = 0.05)
• Factorial: ! (for example, 5! = 120)
• Power (exponent): either ^ or ** (for example, 3^2 = 9)

Parentheses can also be used to nest sub-expressions. For example:
1 + (2 * 3) = 1 + 6 = 7
(1 + 2) * 3 = 3 * 3 = 9

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Financial Calculations

• fv: Future Value
• pv: Present Value
• pmt: Payment
• rate: Annual Interest Rate
• dur: Duration

Financial functions can be used to calculate annuities that cover any conceivable time period so it is necessary to provide a time unit (otherwise values default to seconds). In the simpler cases, annuity functions require three or four parameters and they are calculated using monthly compounding periods.

Using function fv, we can calculate the future value of a thousand dollars invested for 1 year (either "year" or "yr" can be used as a time unit) at five percent yields 1,051.16 dollars. A time unit must be given, but it can be "month" or "mo" instead or a year, or even "wk" or "dy" if this is what is needed.

Function pv gives the present value. We can calculate the present value of a thousand dollars to be 932.58 if it will be received one year from now given a seven percent annual interest rate.

The pmt (payment) function determines the required regular payment that must be made in order to turn a specified initial value into another after a specified amount of time under a given interest rate. Payments are assumed to be made monthly unless specified otherwise (see below). Here we see that given a five percent interest rate, we can accumulate a thousand dollars starting from zero after a year if we make payments of 81.44 each month.

The rate function gives the interest rate required to turn one amount into another in the specified amount of time. Doubling a thousand dollars within a year would require a 71% interest rate (0.71356). Good luck finding this!

The above financial calculations assume the most common case of monthly periods for both payments and compounding interest. But there are many more possibilities. Interest could be compounded semi-annually. Payments could be made weekly or bi-weekly and either at the begining or at the end of this payment period. Durations could be expressed in months instead of years, and so on. The same five financial annuity functions can handle all these special cases just by using more parameters. When longer expressions are needed, you can select the Financial Annuity panel from the menu in order to use buttons. The advantage of using buttons is that they fill in a set of default values for you in the work area and you can then simply edit the content. This saves a good deal of typing and it provides a reminder of what each parameter stands for.

Clicking the fv button loads the work area with its namesake function. Default parameters are also written except for the initial (present) value. The default duration is one year, annual interest rate is 5%, compounding period in monthly. Two more parameters refer to the regular annuity payment amount and period; these values are set to zero for the amount and the payment period is a month. Using these default values on a thousand dollars gives a formula of fv(1000; 1 year; 5%; month; 0; month). This is equivalent to just fv(1000; 1 year; 5%) as previously used and a 1,051.16 result is obtained as before.

Changing the amount and the default values is simple. Using 5000 as initial value, a duration of 18 months and a payment of 100 each month, we have fv(5000; 18 mo; 5%; month; 100; month) which yields 7,253.77. The same 5000 present value for a year, supplemented with weekly deposits of 100 is expressed with fv(5000; 1 year; 5%; month; 100; wk): we have 10,614.86 by the end of the year.

The other financial calculator buttons provide similar functionality. Default parameters that are relevant to each function are filled in. One final parameter that is rarely used in omitted: a final argument value of 1 can be tagged in to indicate that payments take place at the begining of a payment period instead of at the end (the default case). Refer to the AddUp documentation for details on all financial functions and their parameters.

Binary Calculations

Output of all AddUp calculations is given in decimal unless specified otherwise. To explicitly ask for binary output, start an expression with a prefix similar to what is used in front of binary numbers: use '2 at the start of the line. Notice how the quote is on the left side of the requested output base. This line prefix must be followed by a space to separate it from the expression to evaluate. The expression to evaluate could either contain all binary numbers or a mixture of numbers in various numeric bases, but the result will be presented in the number base that the line prefix specifies. The default base-10 applies when no base is specified.

Complex Number Operations

• Natural numbers. These are the first numbers used by mankind. They are simply 1, 2, 3, and so on. These values were created as mental abstraction that could be used to count items.
• Cardinal numbers. These are merely the natural numbers plus 0 (zero). The concept of zero as a number was once puzzling to some, and it had no representation in roman numerals. Since it stands for "nothing" then why even talk about it? Of course it came to be accepted as an actual number.
• Integers. These are cardinal numbers to which a negative sign can be added to represent values smaller than zero. This concept of "less than nothing" was also initially considered unreasonable, and the term "negative" was coined on behalf of those who rejected it.
• Real numbers. These are integers to which a fractional part (or an additional value shown after a decimal mark) is added to represent portions of a whole. Here again, "less than a complete item" can be seen as a foreign concept in some context. But all modern-day students understand the usefulness of real numbers.
• Complex numbers. These are pairs of real numbers handled as a single unit. One of the two components is simply called "real". The other is called "imaginary" due to the fact that it represents a factor of the square root of -1, a value that cannot be represented using a simple real number. But both components of complex numbers are equally real, and complex numbers play a vital role in modern mathematics.

Operations on complex numbers are done with AddUp in exactly the same way as operations on real numbers. The complex number is simple entered wherever a real number could also be used. For example, multiplying two complex numbers is done with an expression such as "(1; 2) * (3; 4)".

A set of complex number functions is implemented in AddUp to explicitly take advantage of these values. They are:

• arg: Returns the argument (or phase angle) of a complex number.
• conj: Returns the complex conjugate of a complex number.
• imag: Returns the imaginary component of a complex number.
• norm: Returns the norm (or absolute value, or magnitude) of a complex number.
• norm2: Returns the square of the norm of a complex number.
• polar: Returns a complex number equivalent to the specified polar representation.
• real: Returns the real component of a complex number.

Conversions

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Hexadecimal Calculations

Output of all AddUp calculations is given in decimal unless specified otherwise. To explicitly ask for hexadecimal output, start an expression with a prefix similar to what is used in front of hexadecimal numbers: use '16 at the start of the line. Notice how the quote is on the left side of the requested output base. This line prefix must be followed by a space to separate it from the expression to evaluate. The expression to evaluate could either contain all hexadecimal numbers or a mixture of numbers in various numeric bases, but the result will be presented in the number base that the line prefix specifies. The default base-10 applies when no base is specified.

Number Base Calculations

Here are examples of numbers expressed using various number bases in AddUp 2.

• Binary (base-2) number 2'10010110 represents decimal value 150.
• Base-3 number 3'121212 represents decimal value 455.
• Octal (base-8) number 8'567 represents decimal value 375.
• Hexadecimal (base-16) number 16'FA represents decimal value 250.

Since number bases can be mixed at will within an expression, it becomes necessary to determine which base will be used to express the final result. The output of all AddUp calculations is produced in base-10 decimal unless specified otherwise. To show a result in another number base, start an expression with a prefix similar to what is used in front of numbers: start the line with a single quote and then the number of the base. Notice that the quote is on the left side of the requested output base now. This line prefix must be followed by a space to separate it from the expression to evaluate. The expression itself could contain all base-12, or all binary, or all octal numbers, or a mixture of numbers in various numeric bases. But the result will be presented in the number base that is specified by the line prefix. The default base-10 applies if a base is not specified.

Converting numbers from one base to another is easily done. Start the line with a prefix that indicates the desired output base: a single quote, the desired output base value and a separating space. Then write the number that will be converted using any desired numeric base. For example, enter '16 2'10010110 to obtain the hexadecimal (base-16) representation of binary (base-2) number 10010110, which will be expressed as 16'96. You can press Enter again from the result line to convert this hexadecimal value to decimal. Since base-10 is the default, the conversion will be automatic and produce number 150 in normal base-10 if no other base is given.

Octal Calculations

Output of all AddUp calculations is given in decimal unless specified otherwise. To explicitly ask for octal output, start an expression with a prefix similar to what is used in front of octal numbers: use '8 at the start of the line. Notice how the quote is on the left side of the requested output base. This line prefix must be followed by a space to separate it from the expression to evaluate. The expression to evaluate could either contain all octal numbers or a mixture of numbers in various numeric bases, but the result will be presented in the number base that the line prefix specifies. The default base-10 applies when no base is specified.

Prime Number Calculations

Essentially, a prime number calculator can determine if a number is prime or not. It may also be able to find the next prime number after and/or before a specific value. AddUp uses three prime number functions to implement these facilities:

• prime: returns 1 if its argument is prime, 0 otherwise.
• prNext: return the next prime number after its argument.
• prPrev: return the previous prime number after its argument.

Since determining with certainty if a number is prime can be an extremely long process, AddUp has a limit on values that can be examined with these functions (which is still reasonably high).

Rational and Fractional Calculations

The format of the output however is controlled by the current output settings. AddUp offers seven different output formats, including rational and fractional formats. The difference between these is that rational results are given as the ratio of two integer values even if the denominator is greater than the numerator (eg. 3/2) while fractional numbers are given with a whole portion if possible, followed by a fraction where the numerator is smaller than the denominator (eg. 1 1/2). This last format will not show any fraction at all if the numerator is zero.

It is important to realize that although results will be shown as an exact fraction, this fraction is simply the best possible approximation of the result given the specified number of decimals used in the fraction. In many cases this fractional result will be exact and correct. But in the case of irrational numbers (such as mathematical constants 'e' or 'pi') then of course the rational value will be an approximation. The value of 'pi' is 3.1415926535897... Using rational output mode this value will be rounded to 22/7 if a single decimal is specified in the format settings. The ratio will be 355/113, a closer approximation, if three decimals are specified instead.

Scientific Calculations

Available mathematical operators are:

• Addition: +
• Subtraction: -
• Multiplication: *
• Division: /
• Percent: % (for example, 5% = 0.05)
• Factorial: ! (for example, 5! = 120)
• Power (exponent): either ^ or ** (for example, 3^2 = 9)

Parentheses can also be used to nest sub-expressions. For example:
1 + (2 * 3) = 1 + 6 = 7
(1 + 2) * 3 = 3 * 3 = 9

Scientific functions are numerous. They include the mathematical functions that are normally found on a scientific calculator: square root, cubic root, power, exponent, logarithm (in various bases), factorial, etc.

Trigonometric functions are included: sine, cosine, tangent, cotangent, secant and cosecant. As well, inverse trigonometric functions are included: arc-sine, arc-cosine, arc-tangent, arc-cotangent, arc-secant and arc-cosecant.

Akin to trigonometric functions, hyperbolic functions are also provided: hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant. As well, inverse hyperbolic functions are included: hyperbolic arc-sine, hyperbolic arc-cosine, hyperbolic arc-tangent, hyperbolic arc-cotangent, hyperbolic arc-secant and hyperbolic arc-cosecant.

For more advanced math and engineering, complex numbers are natively supported. Complex number functions include: real and imaginary extractors, polar conversion, argument (or phase angle), complex conjugate, norm (or magnitude) and square of the norm. Complex numbers can be used directly in most operations where they would normally apply.

More basic arithmetic functions include: absolute value, inverse, modulo and remainder functions, percent and signum.

Statistical Calculations

Statistical functions include:

avg or mean

Returns the average (or mean) of a list of values. It returns 0 on an empty list, or the average value if the list is not empty. The average or mean is the sum of all values in a list divided by the item count.

avgG

Returns the geometric average of a list of values. It returns 0 on an empty list, or the geometric average value if the list is not empty. The geometric average of a list of values is the result of adding the square of all values (eg. using the sum2 function) and extracting the 'count' root of this sum.

count

Returns the count of values in a list (the total number of items). It returns 0 on an empty list, or the number of arguments if the list is not empty.

gcd or gcf

Returns the greatest common denominator (greatest common factor) of all values in a list. It returns 0 on an empty list, or the greatest common denominator of all values if the list is not empty.

lcm

Returns the least common multiple of all values in a list. It returns 0 on an empty list, or the least common multiple of all values if the list is not empty.

max

Returns the maximum value contained in a list. It returns 0 on an empty list, or the value of the largest argument if the list is not empty.

median

Returns the median of a list of values. It returns 0 on an empty list, or the median value if the list is not empty. The median value is the value that is at the center position of a list after it is sorted. For lists containing an even number of values, it is the average of the values found at the two middle positions.

min

Returns the minimum value contained in a list. It returns 0 on an empty list, or the value of the smallest argument if the list is not empty.

prod

Returns the product of all values contained in a list. It returns 0 on an empty list, or the product of all values if the list is not empty.

sd

Returns the sample standard deviation of a list of values. It returns 0 on an empty list, or the sample standard deviation if the list is not empty.

sdP

Returns the population standard deviation of a list of values. It returns 0 on an empty list, or the population standard deviation if the list is not empty.

sum

Returns the sum of all values contained in a list. It returns 0 on an empty list, or the sum of all values if the list is not empty.

sum2

Returns the sum of the square of all values contained in a list. It returns 0 on an empty list, or the sum of the square of all values if the list is not empty.

var

Returns the sample variance of a list of values. It returns 0 on an empty list, or the sample variance if the list is not empty.

varP

Returns the population variance of a list of values.It returns 0 on an empty list, or the population variance if the list is not empty.

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