Negative Values

by Yves Nievergelt, Governor

Jim Holt introduced his review of Robert Kaplan's book on the history of zero in The Wall Street Journal through the following case study.

'Last year in Centre County, PA, the county commissioners voted to abolish an unpopular tax on occupation by evaluating the tax rate officially at zero. Since this left the local school boards painfully short of income, they sued the county, arguing that zero was not a value. To substantiate their claim, their attorney had a former county assessor try to divide by zero on his calculator. It displayed an E for "error."'

Farther in the book review, Jim Holt's mention that if ab = O then a = O or b = O prompted reminiscences in The Wall Street Journal, notably by Bertrand Horwitz (School of Management, Binghampton University - SUNY), about 'a difficult question posed by a 13-year old doing her algebra homework. "Why," she asked me, "does a negative number times a negative number result in a positive number?" An explanation using the number line proved fruitless.'

Several readers came to the rescue, in particular, David Kanowsky: "If I choose to apply strictly algebraic logic to the question, I can also produce an absurdity. If A and B are two positive numbers, then -A and -B are negative numbers, and their product is

-A x -B = (-I x A)x(-l x B)= -1 (A x B),

a negative number." Andrew G. Neal then expanded on another argument of Bertrand Horwitz's (which had apparently not convinced its intended young audience any better than the number line had), using the distributivity of multiplication over addition on the left and on the right,

(A + B)x(C + D)= (A x C)+(A x D)+(B x C)+(B x D),

and substitutions of sample values, 6, -4, j, -2, to demonstrate that

2 = -6 + (-4 x -2),

and hence cleverly to argue that (-4 x -2) must equal 8 rather than -8.

Explanations of these problems had to wait until the following millennium to appear in The Wall Street Journal, and they overlapped four millennia. The school boards' inordinate difficulties in grappling with the concept of "zero" has indeed been traced back to the Roman Empire. In a separate discussion, initially about neither zero nor negative numbers, but about today's (ir)relevance of Roman numerals, yet another reader -- Steve Haggerty -- reported that

"Of further interest is that the Romans never did grasp the concept of zero (0) or negative numbers. They had no need of either."

Evidently neither had the school boards in 1999 A.D.

Who cares? Do you know anyone outside engineering and the mathematical sciences who has ever had to compute (-1) x (-1)? These questions might not arise in "liberal" contexts, where students do not have to study a topic if they do not want to. Yet answers to these questions might be important to instructors of mathematics, in "liberal arts" (whatever that means) contexts, where students may be denied a degree in ancient history if they do not demonstrate a proficiency with such topics as mathematics and statistics for the liberal arts. The alleged need for such topics could be challenged by a fair cross-section of the population -- assisted by uncountably many calculator-punching attorneys -- claiming that despite such a liberal arts education they still do not know whether zero is a value, and much less how to explain the positivity of the product of two negative numbers to their daughter.

While I cannot resolve these issues -- but perhaps you can -- the issue of today's (ir)relevance of Roman numerals was settled definitely positively by still another reader, A. Cohen, whose one-sentence letter to the editor read:

"Your great story appeared in vol. CCXXVI of the Journal."


A. Cohen,"When in Numerals, Do as the Romans Do, "The Wall Street Journal, Western Edition, Vol. CXLII, No. 30, Friday 8 February 2000, p. A15.

Steve Haggerty, "When in Numerals, Do as the Romans Do, "The Wall Street Journal, Western Edition, Vol. CXLII, No. 30, Friday 8 February 2000, p. A15.

Jim Holt, "Bookshelf: This is Really Something, "book review of Robert Kaplan's "The Nothing that Is"(Oxford University Press), The Wall Street Journal, Eastern Edition, Vol. CXCXXXIV, No. 93, Wednesday 10 November 1999, p. A20.

Bertrand Horwitz, "Go Figure, "The Wall Street Journal, Eastern Edition, Vol. CXCXXXIV, No. 99, Thursday 18 November 1999, p. A27.

David Kanowsky, "Accent the Positive, Not Not the Negative, "The Wall Street Journal, Eastern Edition, Vol. CXCXXXIV, No. 108, Thursday 2 December 1999, p. A23.

Andrew G. Neal, "Positively Negative, "The Wall Street Journal, Western Edition, Vol. CXLI, No. 112, Wednesday 8 December 1999 , p. A23.

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Last updated 22 July, 2000