Strange math syntax in old basic listing

I'm browsing through an old book "Basic Computer Simulation" from 1983. It contains a line of BASIC code that reads:

LET Z1 = M * D1 * (PQ / A) [ 3


I understand all of the code except the "[ 3" I don't ever remember that syntax in any BASIC I learned.

The book calls it a "universal subset of BASIC" and says it has been tested on a TRS-80 Model III. Similar lines of code appear in other listings in the book so I don't think it is just a typesetting error.

Another formula in a different listing is:

LET H = H1 - .5 * G * (T1 - T) [ 2


This is a formula for a falling object.

• It's probably a typo in the book (books published in that era were often set by hand, rather than directly from actual source code). I would guess the [ is supposed to be /, but it's impossible to tell without some context. Commented May 27, 2019 at 23:26
• Other possibilities might be + or - because they're both very close to the [ key on a standard keyboard. If it's for a particular machine that might raise other possibilities though, because a lot of the ones at the time had their own peculiarities of layout. Commented May 28, 2019 at 0:28
• You mention the TRS-80, but en.wikipedia.org/wiki/TRS-80_character_set indicates that it didn't even have square bracket characters at all. Its character set had arrowheads in the place normally occupied by square brackets in ASCII. Commented May 28, 2019 at 2:19
• @GregHewgill I think you're on to something. vavasour.ca/jeff/trs80.html has a document that says "note that the [ represents an up-arrow on the TRS-80." and bitsavers.trailing-edge.com/pdf/dartmouth/BASIC_Oct64.pdf says the up-arrow is the exponential operator. Commented May 28, 2019 at 2:48
• @GregHewgill There is enough context from the second equation which is the gravity specific version of s = ut + 1/2at^2 to tell that the [ is exponentiation or ^ in most Basic's. Commented May 28, 2019 at 8:41

Your mention of TRS-80 provides a clue. In the TRS-80 character set, the space normally occupied by the ASCII [ character is instead a ↑ (up arrow) character. Old versions of BASIC (such as this one from 1964) use the up arrow character to indicate exponentiation, probably because at that time the ^ character was not even in the ASCII standard. (There is another question on this site Why do we use caret (^) as the symbol for ctrl/control? which addresses the evolution of ^.)

Your second example LET H = H1 - .5 * G * (T1 - T) [ 2 provides another clue, as it is a formula for position of a falling object, which has a square in the time term.

So, in this book, the [ character represents the ↑ up arrow character because [ in ASCII occupies character number 91, the same as ↑ in the TRS-80 character set. (Note that this differs from the original 1963 ASCII, where ↑ was character number 94.)

When the book was typeset, there may have been a character set conversion problem where this discrepancy was not noticed, or for some reason the publisher simply could not print the ↑ character.

• Pedantic note: by the time the TRS-80 came along, the ASCII standard was well and truly standardised, and the ^ was there. Computer manufacturers had a tendency to call their character set ASCII though, even if it wasn't. Commented May 29, 2019 at 6:23
• Wow. I coded on that platform and didn't realize that ^ was exponent. Commented May 29, 2019 at 20:35
• Note that codepoints 91 and 94 are both in ranges reserved for national variants in ISO 646; they happen to be [ and ^ in US-ASCII, but not in all kinds of ASCII. Commented May 30, 2019 at 20:30
• @TobySpeight There's a non-US American Standard Code for Information Interchange?
– cjs
Commented May 31, 2019 at 3:45
• @Curt, yep - in fact several. It does sound odd (even if you interpreted "American" more widely than "USA", which would be mistaken here). ASCII was invented and named in the US; possibly some of the national variants were too, and ISO adopted it as an international standard. It's a very large topic for a comment; I suggest you ask a question if you want more (and more informed) history on this! Commented May 31, 2019 at 7:22

In old computer books of cheaper sort, (Paperback or pocket books) it was quite common that they couldn't type set all special characters directly. Either they did as in this case, changed the character for something the computer in question didn't use, but the type setter could handle. In this case this is normally mentioned in the foreword of the book. Otherwise you had someone add the character manually to the text. I've seen both versions, even in official manuals...

In this case the [ was a substitute for arrow up, or "to the power of". This is clear from the second example in the question.

• The first version of the ASCII standard (in 1963) had up-arrow and left-arrow characters. Later these were changed to caret (^) and underscore (_) respectively. If you see an old listing with up-arrows, substitute carets when using more modern systems. Commented May 28, 2019 at 13:26
• @KenGober: I suspect the change may have been done because the caret and underscore could serve the same purposes as the original characters (the back-arrow was used to mark a spot where a character had been deleted) but could serve additional purposes as well. Commented Apr 9, 2020 at 19:38

It is the exponentiation operator.

Why is that?

On a TRS-80 Model III the line of BASIC entered would literally be:

LET Z1 = M * D1 * (PQ / A) [ 3


To get the [ character you would press the up arrow key.

You would press the same up arrow key on its predecessor, the TRS-80 Model I. However, it would display as:

LET Z1 = M * D1 * (PQ / A) ↑ 3


This is because the Model I's character set had a few differences from ASCII. Where ASCII has the [ \ ] ^ characters the Model I has ↑ ↓ ← →.

You might wonder why the Model III didn't use ^ for exponentiation. The main reason is that it would prevent compatibility with the Model I which was a big feature of the Model III (even though there were some incompatibilities). I suppose it is possible that the Model III could have accepted both [ and ^ for exponentiation but then those Model III BASIC programs wouldn't be backwards compatible with the Model I.

I also speculate that Tandy would not have been comfortable making the change to the BASIC ROM themselves and would not have been interested in paying Microsoft to do it.

• Interesting note on the difference between the Model I and Model III. Commented May 28, 2019 at 18:59
• Was there a Model II? Commented Mar 3, 2022 at 20:50
• There was a Model II. While it did use a Z-80 it was an entirely different architecture from the Model I, III, 4 line. en.wikipedia.org/wiki/TRS-80_Model_II Commented Mar 3, 2022 at 21:45