You need Microsoft Internet Explorer version 5.0 or better. If you don’t have it, updating to it is painless, both technically and financially. (It is time-consuming through a slow Internet connection, however.) To do it, click here, and then select “Download Now”
Netscape version 6.x is said to be compatible with Microsoft scripting language, but we have not been able to confirm that statement.
A sound card enhances the tutorial. Having 256 display colors or more also helps.
If your screen resolution is less than 1024 × 768, you will have to use the scroll bar a lot, especially during the tutorial.
Netscape version 6.x may have overcome the above-listed limitations. See above.
If you don’t already have a current version of Internet Explorer see above.
This is the down side of the design decision to show menu text as HTML text, rather than in the more conventional menu formats. The up side of this decision is that menu text is much clearer than in a conventional menu, and that menus can show subscripts, superscripts, Greek characters, and mathematical symbols. You can even mark text in the menus by dragging the mouse pointer over it, whereupon right-clicking reveals options that may be useful.
You can click anywhere in an area outlined in magenta. The background changes when the mouse passes over such an area, to point out this fact. Also, following the usual convention, The Universal Units Calculator shows on the Status Line (on the bottom frame) the effect, if any, of clicking the mouse at its current location.
You can click on the text (where the cursor takes its text-editing form), or in the space (where the cursor takes its pointing form). The Universal Units Calculator will correctly select the item in either case.
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This happens if The Universal Units Calculator is used from a site where its use is not licensed. Bluntly, some subhuman has tried to steal it. To use The Universal Units Calculator responsibly and respectably, click here.
A determined criminal can probably steal this program anyhow. If such a person applied his enviable ability constructively, he could have a lot of money, and his self-respect too.
While you are online, Click here, then click on “Update units.” and follow the accompanying instructions.
Currency exchange rates are updated irregularly, but generally every few days. Other units, such as the Speed of Light, seem to have held fairly steady lately.
A non-zero result can on rare occasions be displayed with a zero digit at the beginning, if the number of digits being displayed represents a precision greater than the precision with which the numbers are carried in your computer. If you display that many digits, then not all of the digits displayed are meaningful. The number of digits carried in your computer may vary depending on your hardware (processor type), operating system, Web browser, and other software in your computer.
Specifically, what happens is that a particular number exceeds the radix of the display and yet, when that number is divided by that radix, the result is less than 1. Such a situation is impossible in theory, but can happen in practice because of roundoff error.
If you are displaying only a few digits when this happens, please record exact details and tell us.
An RPN calculator has an effective method of storing and accessing the numerous quantities that are used in and created by an arithmetic process. It is based on an astonishingly simple concept, a “stack” of registers. This term comes from the similarity to a stack of dishes. The uppermost dish is the one that is normally removed for use, and a dish that is added to the stack normally goes on top.
In an RPN calculator, the standard is that the newest entry is shown at the bottom of the stack. When an operation uses one or more quantities as operands (for example, two numbers for addition), it takes the appropriate number of newest entries (for example, the two bottom ones for addition), uses them, and deletes them. The result(s) of the operation are added to the stack as the newest entry(s).
Upon first using an RPN calculator, a user is astonished to find that the requisite operand(s) at each point in the calculation is/are almost always in position ready for use. This is no accident, and indeed almost all computer programs work on the same principle. The reason is a fascinating study in Computer Science, but we recommend that you just go with the flow, and use The Universal Units Calculator’s exclusive register description facility to see that the operands you want are where you want them when you want them.
The Universal Units Calculator has extra controls within the stack, for the rare occasions when operands do not appear in the right places. These take the place of the “memory” registers in other RPN calculators (and of “static variables” in computer programs), and will occasionally be useful.
The term “Polish” reflects that the stack method of accessing what you want when you want it was invented by a Polish mathematician, Jan Lukasiewicz (1878-1956).
The result of an inverse trigonometric function is expressed in the same units as the last argument that you gave to a direct trigonometric function. You can switch at any time either automatically, or by taking a trigonometric function of the desired unit and immediately deleting it.
No matter what the units are, you can freely use the result in operations with other quantities. If a result refers to an angle, it has the same numerical value as if it were in radians.
The units of circular trigonometric functions are determined independently of the units of hyperbolic trigonometric functions.
The display characteristics of the results of inverse trigonometric functions are determined in the same way.
You can achieve the effect of the atan2 function with the two-dimensional polar operation, à(r, θ). It will yield also a “radius”, but you can delete that with one click. Before you delete it, consider. Surprisingly often, you will be able to use it.
For two quantities to form a meaningful total, they must represent the same physical phenomenon, even though they may be of different amount and in different units. When The Universal Units RPN Calculator adds two quantities that form a meaningful sum but are in different units, it expresses the result in the units of the x register. This convention is chosen so that, if you want to convert units, you can conveniently add zero in the units you want.
Logarithms of quantities with units occur meaningfully. For example, the electrostatic potential near a long, charged cylinder is proportional to the logarithm of the distance from the cylinder. However, the value that one calculates from such a formula depends on the units of length that one uses.
A meaningful physical quantity based on a potential is always the difference between potentials at two points. Such a calculation will include a term of the form
| log (r2) - log (r1), | (1) |
which is equal to
| log (r2 / r1). | (2) |
Therefore, in the final analysis, the units of length cancel out.
This cancellation is not good enough for The Universal Units Calculator. Logarithms of quantities with units can occur as intermediate quantities in calculations similar to expression (1). The aim of The Universal Units Calculator is to keep proper track of units throughout any meaningful calculation. In keeping with this aim, The Universal Units Calculator keeps track of logarithms of units.
The basic principle of dimensional analysis is that a unit is a factor in a product. For example, 5 Centimetres is the quantity 5 multiplied by the quantity “Centimetre”. The property of logarithms that
| log (a b) = log (a) + log (b) | (3) |
can therefore be generalized to the statement that
| log (5 Centimetres) = log (5) + log (A Centimetre). | (4) |
A quantity such as equation (4) is not a meaningful final answer, because the quantity log (a centimetre) does not have a defined numerical value. However, when a quantity such as expression (1) is evaluated, the quantities log (a centimetre) in each term correctly cancel, and the result is the same as expression (2). If the calculation is performed incorrectly, yielding a result that is not equivalent to expression (2), the user will be warned by the fact that the quantity log (a centimetre) occurs in the supposedly final result.
No matter what the units of the argument of a logarithm are, The Universal Units Calculator expresses the function value using the logarithms of S.I. units, so that cancellations occur correctly and for clear reasons. Thus, for example, rather than use expression (4), The Universal Units Calculator will use the evaluation
|
(5) |
Thus, if units in the arguments of logarithms do not correctly cancel out, the problem itself and the reason for it are clear.
The Universal Units Calculator recognizes six root “physical phenomena”:
Many operations in The Universal Units RPN Calculator require that operands represent the same physical phenomenon. This does not mean that they have to be in the same units. For example, the sum of two lengths is a meaningful quantity, even if the two lengths are expressed in different units. However, the sum of a length and a time is not meaningful. (Multiplying the time by a speed would make it another length, and then a sum would be meaningful.)
The weight of an object, when it is resting on Earth’s surface, is the force with which it presses against the surface. When you hold the object, you feel that weight as a downward force against your hands. That force depends not only on how much material is in the object, but on the local gravitational field, and your motion (or lack thereof) at the time.
The mass of the object is the measure of the amount of matter in it, independently of the forces acting on it. The above-mentioned units are conventionally units of mass, since they refer to the actual amount of matter. For example, when you buy a pound of butter, you buy a certain amount of butter, not just the quantity that happens to exert a certain force under local conditions. (Selling material by weight, rather than mass (as with balances with springs instead of poises) is illegal almost everywhere.)
The weight, when the object is resting on Earth’s surface, is the mass, multiplied by Earth’s surface gravity. The Universal Units Calculator offers Earth’s surface gravity, so you can get the weight by putting that next to the mass. The gravity turns out to have units of acceleration.
What we call a pound (or whatever) of force is its weight when it is at rest on Earth’s surface. We do that, because Earth’s surface gravity is fairly (not completely) uniform. Earth’s surface gravity as offered by The Universal Units Calculator is a conventionally accepted average.
Obsolete engineering units of force are based on the weights of unit masses. They are available in The Universal Units Calculator by juxtaposition of the unit masses with Earth’s surface gravity.
Part of the reason The Universal Units Calculator can aspire to be Universal without being unwieldy, is that it can form many units as compounds of a wieldy set of units and prefixes. The kilogram is not explicitly listed, because it can be formed from the gram, which is listed, and the prefix “kilo”. The square inch is similarly available from the inch and the prefix square. You can combine a quantity prefix with a power prefix, as in a cubic centimetre.
The pound per square inch is available with just a bit more clicking. It is a combination of “pound”, “inverse square inch”, and (for pressure but not for quantity) “gravity”.
When you compose any such compound, you can click on the components of the unit in any order.
In spite of the aim to produce The Universal Units Calculator, we have been a bit selective about units whose use is geographically narrow. Please tell us what unit we are missing, and who uses it.
Solar time is time marked by the apparent movement of the Sun around the sky, as seen from a fixed point on Earth’s surface. It is the time by which almost everyone lives. The length of the day is subject to slight fluctuations with the time of year. All the astronomical periods in The Universal Units Calculator are mean values. The solar day is an average over the year.
The solar hour, minute, and second are the conventional subdivisions of the solar day. Even the solar day averaged over a year is not a precise constant, because Earth’s rotation rate keeps changing. A “leap second” is inserted at irregular times, as necessary, but it is too unpredictable for The Universal Units Calculator to include it.
Sidereal time is time measured by reference to the Universe outside the Solar System, sometimes known as the “fixed stars”. A distant star, as seen from a fixed point on Earth’s surface, will appear to make a complete circle once per sidereal day. Each year, there is one more sidereal day than solar day.
The sidereal hour, minute, and second are the conventional subdivisions of the sidereal day. The sidereal year similarly is the time Earth takes to return to a point on its orbit, as measured against the “fixed stars”. The Sun will appear to be in the same position, in relation to a distant star, after each sidereal year. Like the sidereal day, the sidereal year is not the period by which most people live, as explained in the following.As seen from a fixed point on Earth’s surface, the Sun moves north and south on an annual cycle, which produces the seasons of the year. The period between vernal equinoxes is called (what else?) the vernal equinox year. It differs from the sidereal year, because Earth’s axis of rotation moves in relation to the rest of the Universe, making a complete cycle about every 25,800 years (known as a Platonic year or great year). Over that period, there is one more vernal equinox year than sidereal year.
The vernal equinox year varies by roughly ±10 seconds during the Platonic year, for the same reason as the solar day varies during the vernal equinox year. Astronomers, and The Universal Units Calculator, use a mean value of the vernal equinox year, known as the tropical year.
Since the tropical year is the year by which almost everyone lives, calendars represent approximations to it. They assign different numbers of days to some years than others, since a tropical year is not an integral (or even a fixed) number of days. The mean Gregorian and mean Julian years are the long-term average lengths of the year according to those calendars. The Julian calendar is out of step with the tropical year by about one day every 131 years. It has been almost entirely superseded by the Gregorian calendar, which is out of step with the tropical year by about one day every 4000 years.
When imperial units were used outside the United States of America, almost all the nations using them defined the inch as 25.4 millimetres. However, until 1958, the United States of America defined the inch to be 1 metre / 39.37. That year, the U.S.A. changed its inch to match that which was in use elsewhere. However, the U.S. government decided not to change all the land surveys that had been completed. Instead, the former inch remains the standard for land surveys in the United States. The irony is, that now that no other nations use Imperial units, the United States could use whatever inch it wants.