Monday, February 22, 2010

The Value of the Leading Zero in Decimals and Writing Numbers

In “getting past the blame – How Can We Save the Next Victim?” (The New York Times Magazine, June 15, 1997; as of 2010-02-22, available at NY.Times.com’s archive) Lisa Belkin explored the causes of errors made by health-care professionals and institutions and how they are discovered and handled. The main storyline driving this article is a minor typographical error with deadly consequences.

Two-month old Jose Martinez had been born with a treatable hear defect. Digoxin was prescribed and the dosage was calculated to be .09 milligrams, Mistakenly the resident who ordered the medication wrote .90 on the prescription. The medication, in a dose that is 10-times stronger, was delivered to the infant and he died. [Note that the aforementioned .90 is the correct one, as reported in the correction following the end of the article, not the number used in the body of the article.

I believe that if leading zeros in decimal numbers were routinely used and, more importantly, expected, that error would have had a better to be discovered and corrected before any harm was caused. If this was the common practice the calculated dose would have been 0.09 and the erroneous prescription would have shown 0.9. The correct number has two '0's before the '9', not one as the wrong one, and is longer. These two discrepancies are easier to spot then the two-digit transposition, which was the actual mistake. If the leading zero in decimal numbers was routinely used and, more importantly, expected, that error would have had a better to be discovered and corrected before any harm was caused.

The discussion, concerning the causes of and for potential remedies for medical errors, should also cover our culture. In particular, the poor habit of omitting the leading zero when writing or saying decimal fractions smaller than one or negative one. All too often people are too lazy to including the zero that preceded the decimal point when no integer larger than zero is part of the number.
That is, when the integer portion of a decimal fraction consists of a zero, we omit it. From a practical consideration, including the leading zero is a wise practice to avoid certain kind of mathematical errors.When the leading zero is routinely used and expected, errors that may result from shifting the position of the decimal point, right or left, are less likely for it is easier to spot them. If the position of the decimal point is shifted right, say from 0.09 to 00.9, two more than one zero appear to the left of the decimal, which is rarely the case. If the right shift is from, say from 0.09 to 0.9, the resulting erroneous number is one digit shorter than the correct one, which is easier to spot. If the decimal is shifted left, say from 0.09 to .009, the leading zero is missing. And if the mistake is, say from 0.09 to 0.009, resulting erroneous number is one digit longer than the correct one; another error that is easier to notice.

At some time subsequently to the publishing of this article, The New York Times published the following, it is at the bottom of the web site of the single-page article:

Correction: The cover of The Times Magazine today and an article on page 28 about medical mistakes render a prescription figure incorrectly. The order for Digoxin given for Jose Eric Martinez, a 2-month-old who died in Hermann Hospital in Houston, was written as .90 milligram (not 0.9 milligram, a different way of writing the same amount). The dosage the child was supposed to receive, under the original doctor's orders, was .09 milligram. The 0.9 rendering was supplied by the Houston hospital; the error was discovered after the article had gone to press.


The error the hospital made reporting the way the dosage number had been written to The New York Times is of another type. This kind of error is also less likely if a leading was expected. The reason for the mistake is this: it is easier to overlook two-character transposition such as from .90 to 0.9. If a leading zero was routinely used and expected, the correct number would have been 0.90 and the erroneous one, being 0.9, is one zero short, a discrepancy that is easier to notice. The result of this type of error is not a change in value; both 0.90 and 0.9 have the same magnitude. However, they indicating different precision: the precision of 0.9 is only accurate to 0.1 (one tenth), while that of 0.90 indicates precision to 0.01 (one one-hundreth), the later is 10 times more precise than the former.

This second error, which is corrected above, confirms my point. It too would have had a better chance to be noticed if a leading zero were included. The reason is this: it is easier to overlook two character transposition such as from .90 to 0.9. If a leading zero was routinely used, the correct number would have been 0.90 and the erroneous one, being 0.9, is one zero short, a discrepancy that is easier to notice.

Trouble is that most mathematics textbooks routinely use this shorter notation — this is the wrong way to save ink or paper — thereby introducing and reinforcing a poor mathematical habit.

Including leading zeros does not eliminate these sorts of errors but it makes them easier to spot.

Writing numbers. But the problem runs deeper. We accept as the norm of the unintelligible chicken scratches physicians scribble on medical charts and prescriptions; we’ll never tolerate such error-prone writing in our financial records and transactions. Prescriptions ought to be treated as checks and draft orders and medical charts as banking records and statements. Drugs should be treated as and given the same respect we afford money. If the standard for writing drug prescription, especially for certain class of dangerous drugs, were at least as formal and at best as rigorous as the checks-writing standard, a whole class of medical errors will be significantly diminished.

It is also a common practice to write financial figures in correspondence using both numerical and English expressions. It is far easier to detect a discrepancy between the two forms of the same number. And it is unlikely that the resident who wrote the prescription for Jose Martinez would have made exactly the same mistake in both formats, writing both the erroneous .09 milligram and its matching error point zero nine milligram instead of .9 milligram and zero nine milligram. (In this case I intentionally omitted the leading zero to show that this method helps detecting errors independently of my leading zero concern.) The fact the two quantities do not match would have been more likely to alert someone to the error. Another advantage of writing health-related numerical quantities in both formats is that it reduce the opportunities to make errors in which the wrong units are written, say using milligrams instead of micrograms or vice versa (which was not an issue in this case.)

It is important to note that both remedies suggested here have nothing to do with medical procedures. The net effect is that anyone, including patients, their family members and other lay people, should be able to recognize such simple errors and alert the professionals.