UCalc - the calculator that includes units by performing dimensional analysis

This shows the main screen of the uCalc calculator showing the units and constants available.

You can run a web version of the calculator by clicking on the above image. But be prepared for a bit of a shock. You must include your units where necessary. For example: 'sin(30)' will not work, you need to include the unit for degrees. This is the small 'o' on the bottom right panel.

The web version is free for you to use in any WebGL compatible browser. The Windows version can be purchased and has many more facilities than the web version, such as being able to save and load formulae, and being able to define your own constants that can then be accessed by a menu.

Current scientific calculators only do half the job, they calculate the numbers whilst leaving out the units. This calculator does both and it does it in a very easy to use intuitive way. The calculation shown above) for the force on a 10kg mass at the Earth’s surface is simply calculated by using the usual keystrokes with a few extra to specify the units. For example:

(1) Clicking on G for the Gravitational constant in the middle left panel followed by the usual multiplication key.

Inputting the mass of the Earth numerically and then the units (kg from the units panel) followed by the usual multiplication key again.

(2) Selecting the digits one and zero from the number pad and then the units (kg) from the units panel

(3) Selecting the division operator from the Operators panel on the right.

(4) Selecting an open bracket from the Operators panel.

(5) Inputting the radius of the Earth ffollowed by the unit (m for meters) and a closing bracket.

(6) Clicking on the up arrow to raise the denominator to a power and then clicking on the number 2 followed by a closing bracket.

(7) Finally the equal sign is clicked on and the/answer is calculated and displayed with the correct units!

Introduction to dimensional analysis

By analysing the fundamental dimensions, such as mass length and time, of each factor in a scientific equation it is possible to determine what dimensions the result of such an equation will have.

In practice this means calculating with the units for these dimensions, such as 'kg', 'm', and 's' which are the shortened form of the SI units for kilogram, metre and second.

In UCalc whenever you enter a number you can follow it with the units for that number.

For example: you can add 10m to 21m to get 31m, but you cannot add non-compatible units. It makes no sense to add seconds to kilograms and the calculator will complain if you try. This is where much of the power of UCalc resides, it makes sure that your calculations make sense by checking that the units are used within an equation in a compatible way

Some units are derived from other units. For example the Newton, N, is the unit of force which is actually kg m s^(-2). Which can easily be verified from Newtons law:

Force (N) = mass (kg) x acceleration (ms^(-2)).

If you were to enter an equation that calculated a mass times an acceleration like this

5 kg x 12 m s(^-2)

You would get the answer: 60 N.

This is because UCalc knows about derived units and will use them when it can to shorten results and hence give you the most useful answer.

NOTE: Units such as metres per second must be entered as ms^(-1) and not as m÷s. The normal multiplication operator cannot divide a pure unit into a number qualified by a unit. (However, this m÷1s would work).

Including the units within an equation is the recommended way to go. It provides a deeper understanding of the equations and eliminates errors. Enlightened textbooks, such as Serwey and Beichner\s (Physics for Scientists and Engineers), shows all its worked examples with units included for each number in each calculation.</p> <p>Consistently use units in all your calculations and you will become much more accurate and efficient in your workings. But now, it is no longer a chore as UCalc will do the work for you.

Using units in your equations

In the following a button is indicated by square brackets. For example: [Hz] indicates the button marked with the label 'Hz'.

To include a unit with any number press the key for the required unit after you have entered the number.

For example: you might want to enter the weight of an item as 12.7kg. Enter the number in the usual way and then click on the units button [kg].

If more than one unit applies to a number then simply follow one unit by another. If a unit is to a negative power then follow the unit by the negative power that applies.

For example you might want to enter a speed of 33.5 meters per second. Enter the number in the usual way and then click on [m] followed by the units [s] followed by the [^] key, followed by [-] and [1] and finally [)].

Using constants in your equations

In the following a button is indicated by square brackets. For example: [Hz] indicates the button marked with the label 'Hz'.

To include a constant press the key for the required constant.

For example: you might want to enter the constant 'pi', simply press [π]. What will be displayed in the top evaluation line is the symbol for pi. When you evaluate the equation the symbol \'π\' will be replaced with the actual value.

Note that pi is dimensionless, i.e. it has no units as it is a simple ratio. This is not true of most constants.

For example: to find the mass of a carbon atom we could evaluate 6 x (mp + mn + me), i.e. six times the combined mass of a proton, a neutron and an electron. This can be achieved with the following button presses:

[6] [x] [(] [mp] [+] [mn] [+] [me] [)] [=]

Notice that the result has the units 'kg'.

Using Memory registers in your equation

In the following a button is indicated by square brackets. For example: [Hz] indicates the button marked with the label 'Hz'.

The memory buttons: [M1] through [M4] can be used to store values that can be used in your expression.

For example I could have an expression that calculates the distance that an object has fallen under gravity for a given time. I might want to
calculate this distance for a range of different times. In this case it would make sense to put the time in a memory location and for the formula to refer to this memory location. I would use the formula g/2 x t^(2), like this.

Enter the equation, (gn/2) x M1^(2), using the key strokes:

[(] [gn] [÷] [2] [)] [x] [M1] [^][2][)]

Click on the text box to the right of [M1] and enter a value of say 10 seconds using the keystrokes: [1] [0] [s].

When you press [=] you will get the result: 490.3325 m.

To repeat the calculation for a different time it is only necessary bring back the formula using the down arrow key and change the contents of the M1 register and press [=].

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