*Prefatory note: The material below apart from that added on March 6, 2013 also appears as Section C.2 of the **Times Higher** sub-page of the **Vignettes** page of jpscanlan.com. The Times Higher page principally addresses the frequency with which observers, including researchers in the world’s leading medical journals, describe, for example, a rate of 3% as “three times higher” than a rate of 1%. As anyone approaching the matter with care would realize, 3% is “three times as high” as 1%; 4% would be “three times higher” than 1%. The material below discusses the fact that most dictionary definitions, if followed, would result in a situation where multiplying 1 by 3 would equal 4. The material is repeated here because it involves a matter warranting attention in its own right, though its reflection on the difficulties encountered by most people in understanding and communicating elementary mathematical concepts is best understood in the context of the other issues addressed on the Times Higher sub-page.. *

**Dictionary Definitions of Multiplication**

On the web site freedictionary.com, one finds three mathematical definitions for the word “multiplication” (from the *American Heritage Dictionary of the English Language*, Fourth Edition (2000), the *Collins Essential English Dictionary* 2d Edition (2006), and the *American Heritage Science Dictionary*, which are set out as items 1-3 below. Dictionary.com sets out, in addition to two of the three that are found on freedictionary.com, three additional definitions (from the *Random House Dictionary*, *Webster's Revised Unabridged Dictionary (1996, 1998), and* MICRA, Inc., WordNet (2006) (of Princeton University)), which are set out as items 4-6 below.

It will be observed that all but the Princeton definition in some manner define multiplication as the process of adding a number to itself a certain number of times. All at least imply that that number of times is the number by which the original number (multiplicand) is then multiplied (what is commonly called the multiplier, though none of the definitions uses the word). Items 3 and 4 are explicit in this regard – that is, in stating that to multiply x by y means to add x to itself y number of times. The former states, for example, that “multiplying 6 by 3 means adding 6 to itself 3 times.” Apparently this definition has been around for some time. A 1970 version of *Webster’s Seventh New Collegiate Dictionary* also states that multiplication involves adding a number to itself a certain number of times.

But each definition employing such usage is patently incorrect. To multiply x by y is either to add x to 0 y number of times or to add x to itself y-1 number of times. Applying the definitions would, as in the case of the example offered by the *American Heritage Science Dictionary*, cause 6 times 3 to equal 24 rather than 18.

*[The following eight paragraphs were added March 6, 2013]*

** **

In February 2013, Australian mathematics teacher/innovator Jonathan Crabtree brought to my attention some surprising examples of the uncritical use of the standard definition of multiplication in circumstances where the definition seems to be receiving close scrutiny. The first involves an October 12, 2012, *Scientific American* item titled “Commuting: Steven Strogatz Explains One of the Laws of Multiplication [Excerpt],” which was the reprinting of an excerpt from the 2012 book,* The Joy of X: A guided tour of math, from one to infinity*, by Steven Strogatz, a professor of applied mathematics at Cornell. The discussion in the excerpt presumably not only received careful attention from Professor Strogatz and the editors of the book, but from the editors of *Scientific American* who determined that the material warranted reprinting.

The discussion, while not questioning the accuracy of the definition, does question its utility for demonstrating commutativity – i.e., that the changing of the ordering of the terms in a mathematical operation does not change the result. In what Professor Strogatz describes as revisiting multiplication from scratch, he observes:

“Take the terminology. Does “seven times three” mean “seven added to itself three times”? Or ‘three added to itself seven times’? In some cultures the language is less ambiguous. A friend of mine from Belize used to recite his times tables like this: ‘Seven ones are seven, seven twos are fourteen, seven threes are twenty-one,’ and so on. This phrasing makes it clear that the first number is the multiplier; the second number is the thing being multiplied. It’s the same convention as in Lionel Richie’s immortal lyrics ‘She’s once, twice, three times a lady.’ (‘She’s a lady times three’ would never have been a hit.)

“Maybe all this semantic fuss strikes you as silly, since the order in which numbers are multiplied doesn’t matter anyway: 7 × 3 = 3 × 7. Fair enough, but that begs the question I’d like to explore in some depth here: Is this commutative law of multiplication, a × b = b × a, really so obvious? I remember being surprised by it as a child; maybe you were too.”

Professor Strogatz goes on to describe what he regards as a more intuitive approach to appreciating the commutative aspects of multiplication.

For instant purposes, however, the interesting aspect of the discussion is that no one involved with creating or reviewing it recognized that “seven added to itself three times” is not the same thing as “three added to itself seven times.” The former is 28 and the latter is 24. Similarly, if one accepts the standard definition reflected in quoted discussion concerning seven times three, a times b does not equal b times a. The former equals ab+a, while the latter equals ab+b.[1]

The second item brought to my attention by Jonathan Crabtree involved an IBM patent for a “high speed multiplier for fixed and floating point operands.” The description of the invention states: “Arithmetic multiplication is defined as a repetition of additions. The multiplicand is added to itself the number of times specified by the multiplier.” I assume that the invention nevertheless causes 7 times 3 and 3 times 7 both to equal 21, though one must wonder how the creation of the coding to do that failed to cause the error in the definition to be revealed. One must also wonder whether in the manufacture of common handheld calculators, and possibly more sophisticated devices, some prototypes were created that actually implemented the definition. Mr. Crabtree advises that many patents reflect the incorrect definition.

Of course the persistence of the definition itself, including the failure to recognize the error in the definition even in the instances below where authors troubled to create putative illustrations of the definition, is itself remarkable enough.

**Definitions**

1. *American Heritage Dictionary of the English Language*:

3. Mathematics

a. The operation that, for positive integers, consists of adding a number (the multiplicand) to itself a certain number of times. The operation is extended to other real numbers according to the rules governing the multiplicational properties of positive integers.

b. Any of certain analogous operations involving expressions other than real numbers.

2. *Collins Essential English Dictionary*

1. a mathematical operation, equivalent to adding a number to itself a specified number of times. For instance, 4 multiplied by 3 equals 12 (i.e. 4+4+4)

3. *American Heritage Science Dictionary*

1. A mathematical operation performed on a pair of numbers in order to derive a third number called a product. For positive integers, multiplication consists of adding a number (the multiplicand) to itself a specified number of times. Thus multiplying 6 by 3 means adding 6 to itself three times. The operation of multiplication is extended to other real numbers according to the rules governing the multiplicational properties of positive integers.

4. *Random House Dictionary*

2. Arithmetic. a mathematical operation, symbolized by a × b, a · b, a ∗ b, or ab, and signifying, when a and b are positive integers, that a is to be added to itself as many times as there are units in b; the addition of a number to itself as often as is indicated by another number, as in 2×3 or 5×10.

5. *Webster’s Revised Unabridged Dictionary*

2. (Math.) The process of repeating, or adding to itself, any given number or quantity a certain number of times; commonly, the process of ascertaining by a briefer computation the result of such repeated additions; also, the rule by which the operation is performed; -- the reverse of division.

6. WordNet® 3.0 (© 2006 by Princeton University)

3. an arithmetic operation that is the inverse of division; the product of two numbers is computed; "the multiplication of four by three gives twelve"; "four times three equals twelve."

[1] Eight readers commented on the item shortly after it appeared. None recognized the problem with the definition. It would seem hard to imagine that none of the magazine's 476,000 plus readers recognized the error, though no harder to imagine than that the incorrect definition would persist in dictionaries for centuries.