Signed numbers, Negative numbers

Negative numbers create problems for many math learners, and really, for good reasons.  We don’t really encounter negative numbers in real life that often.  If we somehow  bounce a check,  buy something that declines in value, or suffer a loss of income, then we encounter negative numbers in “real life.”   If we  live in or travel to parts of the world with very cold weather, say Northern Minnesota in the winter, we will encounter minus degrees of temperature.  Under those conditions,  negative numbers do play a role in everyday arithmetic in addition and subtraction.  But the process and rules for multiplying and dividing negative numbers  would best be left until the subject of  equations in algebra comes up.  Lancelot Hogben made that point back in the 1930’s but evidently nobody listened to him.

Hogben  said  that  negative numbers should  not to be confused with the actual process  of subtraction.  He shows that the rules for negative signs comes from algebra and geometry and its purpose is to balance equations.  That’s why, in algebra, multiplication and division of an odd number of negative signs yields a negative answer and an even number of negative signs yields a positive answer.  Hogben says, “…neither temperature scale nor double entry book-keeping necessarily confers an intelligible meaning on what we learn in school as the rules of signs…”

This is why Kenneth Iverson uses the “high minus” in APL and the underline in J instead of the minus for negative numbers.  He was influenced by Hogben’s book and sought to improve math teaching in both programs.  The underline in J is shown in the examples below:


       _6  +  _4


        _6  +  4


        _3  +  5



        _3  –  5


       _3  –  _5


Scan Addition

       +/_6  _4  _3  _5


        +/_6  4  _3  5


Scan Subtration

        -/_6  4  _3  5


Scan Multiplication

        */_6  4  _3  5


        */6  4  _3  5



        18  %  _3


       _18  %  _3


If you don’t understand the rules for signed numbers, I can do no better than Amby Duncan- Carr at  She takes you step by step through the process for adding, subtracting, multiplying, and dividing signed numbers.  This is probably the simplest and  best explanation of  how to work pencil and paper signed number problems I could find.

I haven’t done so in some time, so I should take this opportunity to link to J Software so that you can get a copy of J if you don’t have it.  It’s free and invaluable.   Choose the link that’s right for your equipment and software setup.

About Richard Rollo

I am a retired Community College Instructor. I taught Political Science 1 American Government for 22 years in Southern California. I am originally from Northern Minnesota. My earliest years were spent in the living quarters of a rural Duluth Winnipeg & Pacific Railway Depot. Then my family joined the great 1950's migration to Southern California where I joined up with fellow baby boomers in overcrowded schools.
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