# Category Archives: Units

Calcpad version 3.0 already supports some non-metric units, that are commonly used in science, technology and engineering. A list of currently supported units is available in Units section of the User manual. More units can be easily appended in the next version. We will appreciate your suggestions very much.

Calcpad supports both Imperal and USCS system. Some of the units have slightly different values although their names are the same. You can select which system to use by switching the respective option in the program window (●Imperial/○US).

You can convert between metric and non-metric units. That is possible because both systems exist simultaneously inside the Calcpad core engine. You can also mix them in one and the same expression. Open Calcpad on your desktop or go to the online calculator and try the following examples:

2ft + 3in|cm

kg/lb

You will get the following answer in the output window:

2ft + 3in = 68.58cm

kg/lb = 2.204623

Converting temperatures is a little bit more tricky because the origins of the different temperature scales are not the same. For example: 0°C = 32°F. Because of that, it makes a difference whether you convert between absolute values or temperature differences. In the next post, we will discuss more about this and how this problem is solved in Calcpad.

# How units work

In this post, we will take a quick look inside Calcpad and see how units actually work. But first, let’s make a brief overview of the theory basis.

## Theory basis

There are only seven base units in the whole universe. They correspond to the seven physical dimensions:

• mass – kilogram (kg)
• length – meter (m)
• time – second (s)
• electric current – ampere (A)
• temperature – Kelvin (K)
• amount of substance – mole (mol)
• luminous intensity – candela (cd)

All other units are derived from the above ones, using the respective laws of physics. For example, forces are measured in Newtons (N). By definition, force is mass × acceleration, so 1 N = 1 kg·m/s2. Such units are called “composite”. They can be always decomposed to their constituent units.

Other method to derive units is by adding prefixes. I this way, we can obtain multiples of both base and composite units. For example,  kilonewtons – kN = 103·N, meganewtons – MN = kN = 106·N and so on. All possible prefixes are listed in the following tables:

Name deca hecto kilo mega giga tera peta exa zetta yotta
Symbol da h k M G T P E Z Y
Factor 101 102 103 106 109 1012 1015 1018 1021 1024
Name deci centi milli micro nano pico femto atto zepto yocto
Symbol d c m μ n p f a z y
Factor 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18 10−21 10−24

However, not all prefixes are commonly used for all units, and that is why they are not included in Calcpad. Full list of the available units is provided in the user manual:

## Internal implementation

Any software implementation must allow the units to be compared, converted and scaled. For that purpose, we need to store the information for how units are composed by the base ones. We also need the corresponding scale factors, in order to allow for multiples. Then, we have to define all possible operations like comparison, addition, subtraction, multiplication, division and exponentiation.

To store all this information, we need a special type of data called “structure”. It can contain multiple values of different types in a single variable. In the case of units, we will include a name and two vectors. The first one will store the exponents for all seven physical dimensions (mass, length, time, current, temp, substance, luminosity) and the second one – the respective scaling factors:

`Unit = {name: "name", exponents: [p1, p2, p3, p4, p5, p6, p7], factors: [f1, f2, f3, f4, f5, f6, f7]}`

A base unit can include only one value of “1” for the respective dimension. All the rest must be exactly zero “0”. Composite units can have several nonzero values for the corresponding dimensions. Bellow are some examples of unit definitions:

Base units:

```Kilogram = {name: "kg", exponents: [1, 0, 0, 0, 0, 0, 0], factors: [103, 1, 1, 1, 1, 1, 1]} Meter = {name: "m", exponents: [0, 1, 0, 0, 0, 0, 0], factors: [1, 1, 1, 1, 1, 1, 1]} Second = {name: "s", exponents: [0, 0, 1, 0, 0, 0, 0], factors: [1, 1, 1, 1, 1, 1, 1]}```

Composite units:

```Newton = {name: "N", exponents: [1, 1, -2, 0, 0, 0, 0], factors: [1, 1, 1, 1, 1, 1, 1]} Pascal = {name: "Pa", exponents: [1, -1, -2, 0, 0, 0, 0], factors: [1, 1, 1, 1, 1, 1, 1]}```

Multiples:

```Ton = {name: "t", exponents: [1, 0, 0, 0, 0, 0, 0], factors: [106, 1, 1, 1, 1, 1, 1]} Millimeter = {name: "mm", exponents: [0, 1, 0, 0, 0, 0, 0], factors: [1, 10-3, 1, 1, 1, 1, 1]} Minute = {name: "min", exponents: [0, 0, 1, 0, 0, 0, 0], factors: [1, 1, 60, 1, 1, 1, 1]} Kilonewton = {name: "kN", exponents: [1, 1, -2, 0, 0, 0, 0], factors: [103, 1, 1, 1, 1, 1, 1]} Megapascal = {name: "MPa", exponents: [1, -1, -2, 0, 0, 0, 0], factors: [106, 1, 1, 1, 1, 1, 1]}```

## Units consistency

You can compare, add or subtract only values, which units are consistent. For example, “kg + m” does not make any sense, because both operands represent different physical dimensions. The expression “m2 + m3” does not make sense as well, because you are trying to add area to volume. However, “m + mm” is possible, because both represent length and we can convert them to each other. So, units are consistent only if they are of the same dimensions and with the same exponents. What we need to do for our structure, is to check if the exponents vectors match exactly.

Even if consistent, units are not necessarily equal, because they can have different scaling factors. So, we still cannot operate directly with the respective values. First, we have to convert all values to the same units. For example:

`3 dm + 2 cm = 3 dm + 0.2 dm = (3 + 0.2) dm = 3.2 dm`

Subtraction and comparison are performed in the same way as the addition. The following algorithm can be used:

1. Check if the units are consistent (compare the exponent vectors). If not, report an error. Otherwise, continue to step 2.
2. Compare the scale factors. If different, compute the conversion factor for the second unit and use it to multiply the second value.
3. Perform the operation and return the result with the units of the first value.

## Multiplication, division and exponentiation

Unlike the previous ones, these operations do not require the units to be consistent. Actually, this is the way we use to derive new units by combining the existing ones. In the software, these operations are performed by manipulation of the respective exponents as follows:

1. Multiplication is performed by addition of the exponents:
`m·m2 = m(1 + 2) = m3`
2. Division is equivalent to exponent subtraction:
`N/m2 = kg·m·s-2/m2 = kg·m(1 - 2)·s-2 = kg·m-1·s-2 = Pa`
3. Exponentiation is performed by multiplication of the old and new exponent:
`(m2)3 = m2·3 = m6`

The following algorithm is applied:

1. Check if there are different scale factors and compute the conversion factor.
2. Perform the required operation (multiplication, division …) with the units and obtain the resulting units.
3. Perform the operation with the values, apply the conversion factor and return the result in the calculated units.

## Target units

If you do not specify any target units, the result will be decomposed to the base units. For example:

`100N·10m = 1000 kg·m2/s2`

That is because the program cannot understand the physical meaning of your formulas. Sometimes, the result can be ambiguous and the program will not know what to do. For the above expression:

• If it is bending moment or torque, the result will be in Nm.
• If it is work or energy, the result will be in Joules.

In such cases, you have to specify the target units and if they are compatible, the result will be converted automatically. Always write the target units at the end of the expression, separated by a vertical bar “|”:

`500N·20m|kJ = 10kJ`

`2000N·5m|kNm = 10kNm`

# Calculating with units

Unit conversion is one of the most tedious jobs when doing calculations. Although not very difficult, it is repetitive and time consuming.

For example, lets calculate the stress in a steel beam. Assume that span length is in meters, section dimensions are in mm, geometrical properties are in cm2 (cm3, cm4), section forces are in kN (kNm) and stress is in MPa. That requires you to include a lot of conversion factors in your formulas. So, it would be very helpful if you could write the units next to the respective values and let the software do all the conversions for you.

The most recent version Calcpad Pro 3 supports physical units in calculations. For now, only metric units (SI compatible) are included, but more units will be added in the next versions.

You can attach units to every value in an expression and obtain the result in the specified target units. The target units are added at the end, separated by a vertical bar “|”.  , Open the online calculator and try the following expression by yourself:

`2m + 50 cm + 200 mm|dm`

Since the target units are “dm“, the answer is 27dm (not just adding 20 + 50 + 200 = 252). That is because all values are converted automatically to the same units before the addition: 20dm + 5dm + 2dm = 27dm. If you do not specify any target units, the units of the first operand will be used.

If you want to do simple unit conversion, just write the value, followed by the source units, a vertical bar and the target units:

`25m/s|km/h`

The result is 90km/h, which is correct.

You can also attach units to variables and functions. If you have units in an expression that defines a variable, the result will be stored in the variable, together with the calculated units. If you specify target units at the end, they will be used instead. From this point further, the variable will be substituted always with the internally stored units. You can try the following examples:

Input Output
Example 1: Speed
```'Distance -'S = 50m
'Time -'t = 2s
'Speed -'V = S/t|km/h```
Distance – S = 50m
Time – t = 2s
Speed – V = S/t = 50m/2s = 90km/h
Example 2: Force
```'Mass - 'M = 500t
'Acceleration - 'a = 4m/s^2
'Force -'F = M*a|kN```
Mass – M = 500t
Acceleration – a = 4m/s2
Force – F = M·a = 500t·4m/s2 = 2000kN
Example 3: Stress in column
```'Column load -'F = 2000kN
'Section size -'b = 500mm
'Section area -'A = b^2|cm^2
'Stress -'σ = F/A|MPa```
Column load – F = 2000kN
Section size – b = 500mm
Area – A = b2 = (500mm)2 = 2500cm2
Stress – σ = F/A = 2000kN/2500cm2 = 8MPa

If you specify target units after a function definition, they will be attached permanently to this function. The result will be converted to them whenever the function is used further. However, you have to be careful to provide the proper units for the arguments. Otherwise, you can get an error.

Although Excel is still the most popular platform for engineering spreadsheets, it lacks such advantages as automated units. I think that math software is more appropriate for engineers in general, but this will be a topic for future discussions.

Read the next post to see how do units actually work. We will lift the cover and have a quick look inside the Calcpad engine, together.