(An archive question of the week)
Last time we ended with questions about writing (or ignoring) zeros at the end of a decimal. I didn’t have room for one more question, so I’ll put it here.
Zeros at the start
This question comes from 2000:
Zero Before the Decimal Point? A fellow instructional designer and I are having a philosophical dispute over the purpose and philosophy of textbooks and teachers showing the zero in front of a decimal number. (I have a teaching certificate and an M.A. in instructional design, while he has a Ph.D. in math.) "0.34 x 0.298 =" or "0.562 / 0.25 =" As a former elementary teacher for 12 years, I contend that the zero is redundant to the decimal, is a harmful crutch, and can even be distracting. The prerequisite skills up to the decimal include mastery of place values, and the comprehension of the decimal point. However, I still see it used in classes past the fourth grade. I agree with him that when used in a text sentence, it makes sense to distinguish punctuation. I also agree that when the decimal point is initially introduced, it can be used as a transition. I disagree with him that when used in a stand-alone math problem, it is still visually pleasing and, I guess, therefore better for the mind to compute accurately. Please explain and arbitrate.
Joe has nicely broken down the issues. In writing, using the zero, as in 0.25, can make the decimal point’s purpose clear, in contrast to periods or other punctuation. But in writing out mathematical work on its own, in print or in handwriting, would it be better to avoid the unnecessary zero and just write this? $$.34\times .298$$
Hi, Joe. My main problem with your reasoning is that you evidently think that redundancy is bad; I think it's good. Take another example, the use of parentheses: There are many situations where we don't need parentheses because the rules make the meaning clear; yet I recommend putting them in, just to make it easier for the reader, and to ensure that if a rule is forgotten, it will be hard to misread it. (This is commonly recommended for good computer programming practice, where there are extra rules that are easy to forget.) Sure, one could use extra parentheses as a crutch, putting them in everywhere to avoid having to learn the rules at all; but that's only in the extreme. Crutches are useful to those who need them.
Yes, overuse of parentheses can be a bad thing, making an expression confusing by distracting the reader; but underuse of parentheses is also problematic. Up to a point, at least, redundant information can be helpful. My own background is relevant: At the time I worked as a software engineer on a project in which redundancy was a selling point, making the system safe against failures.
In the case of the zero before the decimal point, although it doesn't affect the value of the number, it is almost universally considered good practice to include it - not for mathematical reasons, but for human reasons. It can be easy to miss the decimal point, and the zero makes it stand out. It's not essential, but it's a good habit to develop for those occasions when clear communication will be necessary.
Even outside of a sentence, that is, the zero calls attention to the decimal point, so the reader is less likely to overlook it. (This applies both when you are communicating with someone else, and when you are communicating to yourself! You can misread even something you yourself just wrote, in the course of a calculation.) And if we make this a habit even when it probably doesn’t matter, we are more likely to do it when it does matter.
I never like to make decrees on my own, so I checked my claim that my opinion is common. None of the links I gave still work, but I found similar references still available, which I’ll include here.
Just to make sure this was not my own quirk, I did a quick search for other people who might make the same - or opposite - recommendation. Here are a few: First, in writing drug prescriptions: Good Prescribing Guidelines (Westmead Hospital Department of Pharmacy - Khai Bui) http://www.westmead.nsw.gov.au/alliedh/pharm/pages/goodpres.htm "Never leave a decimal point naked, such as .5 mL. When the decimal point is not seen, a tenfold overdose may occur. "When a decimal fraction must be prescribed, always write a zero before the decimal point. "Never put a decimal point and zero after a whole number such as 2.0 mg. This should be written as 2 mg. If the decimal point is not seen, a tenfold overdose may result."
This last example is clearly important in this context, but inappropriate in contexts where significant digits matter, as in engineering. The zero at the end, in such contexts, does mean something, and is not there just to hold a place.
Here is a current source, Today’s Medical Assistant – E-Book: Clinical & Administrative Procedures:
Guidelines for Completing a Prescription Form
4. Medication dosage
- Never leave a decimal point “naked”. If the dosage is a fraction of a unit, a zero must be placed before the decimal point as a means of focusing on the fractional dose. This reduces the possibility of misreading the dose as a whole number. Example: 0.5 mL (not .5 mL).
- Never place a decimal point and a zero after a whole number because the decimal point may be overlooked, resulting in a 10-fold overdose error. Example: 5 mg (not 5.0 mg).
Again, Medicine, Malpractice and the Law (a paper by Raymond Wacks) http://www.medicine.org.hk/bma/programme.htm "The expression of drug dose and units should be clear. For whole numbers it is better to avoid following with a decimal point and a zero which may be misinterpreted as ten or one hundred times the appropriate dose. For numbers less than one it is essential to place a zero before the decimal point."
Here is an interesting current discussion of the real possibility of such errors, from Preventing 10-Fold Dosage Errors in Pharmacy Times:
The risk of 10-fold overdoses is made greater by health professionals and computer systems that dangerously use trailing zeros (eg 1.0 mg, which can be misread as 10 mg) or by health care workers who do not use leading zeros (eg .5 mg instead of 0.5 mg, the former of which can be misread as 5 mg). …
Similarly, in product labels: FDA Guidelines (U.S. Government FDA/ORA Compliance Policy Guides) http://www.fda.gov/ora/compliance_ref/cpg/cpggenl/cpg140-500.html "A zero before the decimal point should be used in numbers between 1 and -1 to prevent the possibility that a faint decimal point will be overlooked. "Example: The oral expression "point seven five" is written 0.75."
This is essentially the same as medical concerns. These guidelines are currently found at CPG Sec.140.500 Metric Declarations of Quantity of Contents on Product Labels (Supplemental Guidance 3G).
In style guides
Next, in the metric system: Metrics the Right Way (George Sudikatus, ICF KH Metric Coordinator, Pacific Northwest National Laboratory) http://www.pnl.gov/ag/usage/metrics.html "In the United States, the standard decimal marker is a dot on the line (i.e., a period or 'decimal point'). When writing numbers less than one, add a zero before the decimal marker. For example, on a drawing you might define a small length in English units as .032 in., but write the metric length as 0.81 mm."
Many people don’t know how much is specified in various government and international rules about the metric system standard! Here, they don’t care how you write inches, but they want metric measurements to be written in a way that maximizes clarity.
A standard U.S. guide for the metric system (from NIST) is Writing with Metric Units:
The dot or period is used as the decimal point within numbers. In numbers less than one, zero should be written before the decimal point. Examples: 7.038 g; 0.038 g.
Also, some publishers and organizations include it as part of their style guides: Instructions for Authors (Publisher: Taylor and Francis Group) http://www.tandf.co.uk/journals/authors/t-authors/wstauth.html "Use a zero before the decimal point for numbers less than one. For example: t = 0.40 "However, do not use a zero before the decimal point when the number cannot be greater than one. This occurs with correlations, proportions and levels of statistical significance. For example: r = .27, p < .01
This is interesting: They imply that when context makes it clear that there will be a decimal point (27 would make no sense for a correlation coefficient), you can leave out the zero – and in fact, should.
The same rule is used in APA Style, here in their Numbers and Statistics Quick Guide:
see Publication Manual Sections 6.36 for guidelines on decimal places
- Put a zero before the decimal point when a number is less than 1 but the statistic can exceed 1.
- Do not put a zero before a decimal when the statistic cannot be greater than 1 (proportion, correlation, level of statistical significance).
But some take an opposite position:
On the other hand (!): Preparing Manuscripts for Demography (Department of Demography, Georgetown University) http://www.georgetown.edu/departments/demography/main/Student/demo.htm "Decimal fractions should not include a zero before the decimal point (e.g., .05 is correct; 0.05 is incorrect)."
They give no reason, and I can’t find any remnants of this opinion now. But I did discover a drafting standard, ASME 14.5, that says (about inch measurements only, not metric),
1.6.2 Decimal Inch Dimensioning
The following shall be observed where specifying decimal inch dimensions on drawings:
(a) A zero is not used before the decimal point for values less than 1 in.
This leaves us with two exceptions to the rule: In statistics, don’t use the leading zero for numbers restricted to the interval \((-1, 1)\), and in drafting, don’t bother with it except for metric measurements.
These are not carefully chosen references, just those that I found in a quick search. They should suggest that inclusion of the zero is a common, though perhaps not universal, practice, and has good reasons behind it. Therefore, I think it is appropriate for students to become familiar with this style. We can let them become lazy later - if they don't become pharmacists. On the other hand, I don't think I would require them to always put in the zero themselves; and I would make sure they saw numbers written without the leading zero to make sure they knew it meant the same thing.
This last pedagogical comment is important. Students should learn this “rule” not as a mathematical requirement (as if .45 did not mean the same thing as 0.45), but as a matter of good style to prevent human error. They will see both forms in the real world, and should be able to read them (without a compulsion to pull out a red pen and write in the zero), even if they rarely write decimals without the zero.
For a concurring opinion from 2005, see: