Money is the 8th dimension. Really. At least in Mathcad.

What Is a Dimension?

In math, dimension describes the number of parameters needed to define a point in space. In engineering, it means any directly measurable physical quantity. These definitions are not incompatible. Both describe something concrete: length, mass, temperature, and so forth. In fact, SI (from the French, Le Système International d'Unités) declares 7 basic dimensions from which all other physical quantities can be derived. As a reminder, they are: Mass, Length, Time, Current, Temperature, Luminosity, and Substance... and Cost!

Cost? While not a physical quantity, organizations that make physical objects are concerned with cost. Engineers build cost optimizations into their solutions. Engineering companies need to track and associate cost with the goods they design and build. So, engineers like to represent cost as a dimension, a tag that endows numbers with physical, or at least intellectual, meaning. That's how Money came to be dimension number 8 in Mathcad. If you look on the Tools > Worksheet Options > Unit System tab, you'll see it listed under Base Dimensions, right in between Mass and Substance.

Each dimension has one or more units that allow you to say "how much" of that dimension you've got. Eight meters means something different from eight inches - it's longer, as indicated by its units - but it means something categorically from eight seconds or eight kilograms. They're different dimensions. This notion is deeply ingrained in engineers' understanding of how to solve problems.

Base Units

But not all dimensions are created equal. Let's examine a few. For example, most dimensions have well-defined base units that provide the multiplicative conversion rates to all other units. If we're talking about mass, SI has the kilogram, CGS has the gram, and the US system has the pound. Each is convertible to other units in its system and to each other by multiplicative scaling factors.

Now, look at temperature. It has a linearly scaled base unit of Kelvin, but to recalculate a value to Celsius or Fahrenheit, you must use functions and inverse functions, and you must also concern yourself with differential temperature versus absolute temperature.

Despite a few untidy unit conversion corners here and there - people haven't debated this stuff for hundreds of years for nothing, you know - it seems that choosing a suitable base unit is key.

Monetary Base Unit

What would the "base unit" of Cost be in any of those systems? We might have used the gold standard. We could have chosen the US dollar or the UK Pound or the EU Euro. If there were a constant conversion rate between any of these currencies and all other currencies, then it wouldn't have mattered so much what we used as the base symbol in Mathcad. But, in fact, the variability of the conversion rates is the difficulty that kept Monetary units from becoming a standard in Mathcad until recently.

In Mathcad 14, we chose a Monetary base unit. It is, in fact, the "Currency Sign" in the UNICODE font set. Insert > Unit > Dimension: Money.

What this means is that you can track the dimension of Money without implying any particular currency or conversion rate. As with all dimensions in Mathcad, you can keep track of it in calculations and show imbalances when other dimensions are added to a monetary calculation by mistake.

Tracking dimensions is one of the primary strengths of Mathcad so this new dimension can be used in all the ways that other dimensions are balanced. Now you can optimize on Money or return units of Money as part of a larger calculation.

Example

Suppose you want to determine the minimum cost-effective order quantity of a material, when you know the annual demand.

total weight of material needed in a year

costs to place an order

cost per unit weight

charge for keeping a unit in inventory

Minimize the total cost of inventory and sales function:

guess value

The business requires a minimum order of about 800 kg at a time that gives a total cost of inventory and sales of

So far so good. It just looks a little unusual. To make this display by default use some other monetary symbol, choose the Tools > Worksheet Options > Display tab, and pick another currency symbol. For example, it's possible to stick within one monetary system and do a variety of conversions, if that's useful to you. You can convert in dollars, if you like, and even define some SI style prefixes for your currency. For example,

Usefulness of Money Dimension

The creation of a new dimension for Money, while not strictly consistent with the intentions of organizations like the SI, gives you a powerful tool for creating your own self-consistent set of symbols and conversions. It allows the exploration of problems with the cost included in the equation.

Even in an era of volatile currency conversion rates, this tool is quite useful for both estimating and updating the cost of projects that involve multiple currencies. Because of Mathcad's ability to recalculate, it's quite easy to create a calculation that shows the cost based on several different currencies, then update the conversion rates to redo the computation.

To change the display of the currency symbol click on it and go to Tools > Worksheet Options > Display Tab > Currency. Then you can pick Euros, Pounds, Yen, or Hong Kong Dollars, as above or other currency units.

The rates can be changed by hand, or read in from a data file. Or you can lock the rate to a particular point in time when the estimate was made.

While the issue of how to represent Money in a dimensional balancing tool that relies on a base value is not quite obvious, the end result is effective. If you ignore what the base is and simply say that there is one, you can track and balance this dimension as well as any other, and worry about applying precise values to the results at the end of the process.

Note: The cost unit was added to Mathcad 14.0 in m010. You can download this free upgrade to Mathcad 14 from http://www.ptc.com/support/mathcad.htm. Users with maintenance contracts can click on the link under Downloads. Single users should click on the Single-User tab, then Download Mathcad Updates below.