At a used bookstore today, I picked up the 1893 text *Elements of Arithmetic: For Primary and Intermediate Classes in Public and Private Schools* by Dr. William J. Milne. One thing that I noticed was that he is adamant that “and” is never to be used when naming integers: “In reading numbers expressed by three *figures,* the tens are read after the hundreds without the word *and.* Thus, 478 is read four hundred seventy-eight, instead of four hundred *and* seventy-eight” (p. 78). That Milne feels the need to emphasize this suggests that “and” was in standard use here, so the he feels the need to “correct” it, at least in rigorous mathematical contexts.

This stood out to me because I don’t recall hearing about this convention when I was in school, but I understand it’s prevalent in elementary school these days. My students have corrected my using “and” in that context, but they weren’t clear on why. It had something to do with “and” only being used in decimals. However, that doesn’t make sense, because the standard way of reading decimals is to use the word “point” (e.g., “pi is approximately three point one four one five nine”). Given the age of the book, I was curious if Dr. Milne had anything else to say on the matter.

He defines mixed numbers on p. 150, but doesn’t explicitly tell how they’re supposed to be read. However, prior to introducing mixed numbers formally, he gives examples such as “1 and 5 sixths” and “2 and a half” (p. 147). This strongly suggests that mixed numbers are to be read with “and” between the integer portion and the fraction.

In the chapter called “Decimal Fractions”, Milne makes this explicit. He describes decimals as being shorthand for fractions: 0.9 represents \(\frac{9}{10}\), while 0.48 represents \(\frac{48}{100}\). “Since decimals have the same law of increase and decrease as integers,” he writes, “the *denominator* of the fraction may be indicated by the *position* of the figures” (p. 188).

When reading the number 796.584694, for instance, Milne feels we should say, “Seven hundred ninety-six and five hundred eighty-four thousand six hundred ninety-four millionths”. Milne writes, “Hence it is evident that: *In reading a decimal, the decimal should be read as an integer, and the denomination of the right-hand figure should be added.* In reading an integer and a decimal, use the word *and* only between the integral and decimal parts of the number” (p. 189).

This approach is naturally problematic when it comes to irrational numbers, which Milne doesn’t address in the book regardless. I reckon he may have gotten around this by describing the functions on algebraic numbers, using the names for transcendentals, and describing the decimal approximations as just that (e.g., “\(\sqrt{2}\pi \approx 4.443\)” would be read aloud as “the square root of two times pi about equals four and four hundred forty-three thousandths”). However, it seems needlessly confining to require someone to decide how many decimal values of π they’re going to give before they’ve started speaking (“three and fourteen-hundredths” vs. “three and one thousand four hundred sixteen ten-thousandths”). And, at any rate, this is confusing to the audience, compared to the standard convention of “three point one four one six”.

While I’m intrigued by Milne’s suggestion that decimal forms are abbreviations for fractions with denominators which are powers of ten, I don’t agree with his method of reading such numbers aloud, nor do I agree that “and” is necessarily wrong in the integer portion. He uses “and” elsewhere for addition (“three and one” etc. on p. 15, for instance). It is true that 400 + 78 = 478, which he would read as “four hundred and seventy-eight are four hundred seventy-eight” (he uses “are” rather than “is” for equality, which is worth a separate post).

Overall, though, I was intrigued to find a convention that I have been identifying as a relatively recent shift in such an old book.

**Follow-up:** It’s been pointed out to me that, in many applied fields, people already know the level of precision they’re using from the outset, so it wouldn’t be as much of a burden, nor would it necessarily be confusing, to always refer to decimals in this fractional form. If everyone involved in a conversation knows, for instance, that all decimal values are being given in thousandths, then it may actually be clearer to say “four thousand thirty and twenty-five thousandths” than “four thousand and thirty point zero two five”, both of which are clearer than the casual speech “forty thirty point zero twenty-five”. H/T Tyler Bartunek and others.

Joel David HamkinsI went to elementary school (USA) in the early 1970s, and the convention about omitting “and” was definitely a standard part of the curriculum.

Pat ballewThe basic idea, and many texts old and new assert this, is that we are reading units AND pieces, just as we give fourteen dollars and fifty-two cents, or five feet and three inches. It is a conjunction joining two different things.

Paul Hartzer(Post author)Fair enough, but the claim is that “and” in “five hundred and three” will lead to confusion. It doesn’t. Everyone knows what “five hundred and three” means. At worst, it will lead to the question “three WHAT?”, the correct answer to which should be “three UNITS” — thus being an opportunity to repeat a clarification that bears repeating regardless.

milo gardnerThe 1890s was a time when Gauss’ 1801 book “discussions on arithmetic” was provided as background to elementary arithmetic books. By 1910 an international convention further placed modular arithmetic as a foundation for college arithmetic courses.

Of course, WW I ended the math fad experiment, with all things German were removed from US, British and French math books. Classical Greek geometry was re-introduced in ways that need to be recalled.

Math fads continue to muddle arithmetic education.

My 1950s high school math education used set theory as background. By the 1960s set theory was moved to the front of the class in another failed math fad, one os several in vogue over thte last 50 years.