Concepts of Zero: A Mathematical Study for Grade 6 and Up

In the “Numbers and Operations Standard” section of the National Council of Teachers of Mathematics’ Table of Standards and Expectations, it states that, in grades 3 through 5, all children should “understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.”[1] Starting in the sixth grade, all children should “develop meaning for integers and represent and compare quantities with them.”[2] The glossary of the Massachusetts Mathematics Curriculum Framework defines whole numbers as “a number that is either a counting number or zero,”[3] and in Standard 4.N.9 requires that students “select, use, and explain theâÂ?¦identity properties of operations on whole numbers in problem situations,”[4] so I decided to focus my investigation on the history of and mathematical thinking behind the concept of zero.

In their article “A History of Zero,” J.J. O’Connor and E.F. Robertson discuss two uses of zero: as a number in and of itself, and as a placeholder. A useful way for remembering the difference between the two uses [5] is that as a number it is used in such operations as 1+0, 0-1, 0/1, and 1×0. As a placeholder it is used in equations like 1+10, 10-1, 1×10, and 10/1.

The method of using a placeholder so that the positions of the surrounding numbers are correct (differentiating between, for example, 64 and 604 and 6004) dates back to the Babylonians around 400 B.C.E., and may have even been found in Mesopotamia around 700 B.C.E. O’Connor and Robertson compare the early use of zero to a “punctuation mark, so that the numbers had the correct interpretation.”[6] The Babylonian place value system was not base-ten (counted in tens) like ours; they used a sexigesimal number system, which is to say that they counted in sixties. When we count the minutes in an hour, we are using a sexigesimal system.

This early zero did not look like our modern notation; the Babylonians used wedge-shaped symbols, and the Mesopotamians used hook-like symbols (see picture at http://www.mediatinker.com/whirl/zero/zero.html).

In the picture, Babylonian cuneiform shows the difference between 64 (1 sixty and 4 ones) on the top, and 3604 (1 “sixty squared” + zero sixties (nothing in the sixties column, denoted by the two smaller superscript wedges) + 4 ones)[7] It is thought that our current representation of zero comes from ancient Greek mathematicians who, in recording astronomical data, used the symbol O (for unknown reasons).

What is known is that around 650 C.E., the use of zero as a number showed up in Indian mathematics, for the first time being used to mean “nothing.” The Indian mathematician Brahmagupta attempted to create rules for arithmetic with zero, negative, and positive numbers. He started by saying that if you subtract a number from itself you get zero, then followed with:

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zeroâÂ?¦.A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.[8] This description matches our modern arithmetic use of zero – the understanding that when it comes to addition and subtraction, zero does not alter the number upon which it operates.

Multiplication and division are trickier. Brahmagupta goes on to say that any number multiplied by zero is zero (which also matches current usage), and that “A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.”[9] Five hundred years later, the mathematician Bhaskara disagreed with Brahmagupta, stating that

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.[10]

Though the depth of these mathematicians’ thinking is apparent and extensive, we now know that it is impossible to divide anything by zero, and zero divided by something else remains zero.

Islamic, Arabic, and Chinese mathematicians picked up the use of zero by the twelfth and thirteenth centuries, and Italian mathematician Fibonacci was one of the primary proponents of it (and the other numbers in the Hindu-Arabic number system) in Europe. He started using it around the year 1200, but it did not become more widely used until the 1600’s (and even then it was met with a great deal of skepticism).

Today, there is still a great deal of debate about what exactly zero is. Is it really a number? The simple answer is yes – it’s the additive identity, or the number that, when added to any other number (x), doesn’t change the value of x. It is a member of the set of real numbers, which is the complete ordered field of numbers. It is also a member of the sets of positive integers (the set of natural numbers and zero) and (all) integers (natural numbers, zero, and negative numbers, such as -1, -2, -3, etc.)). It is not, however, a member of the set of counting/natural numbers, since you don’t start counting with zero. As an interesting side note, negative numbers are also called additive inverses; the additive inverse of a number is something that you can add to the original number to get the additive identity (0). The additive inverse of 1 is -1; the additive inverse of 4 is -4, etc. Also, all numbers but zero have a multiplicative inverse, or a number you can multiply by the original number to get 1 (for example, the multiplicative inverse of 7 is 1/7). There is nothing that you can multiply by zero to get 1 – when you multiply with zero, you always get zero.

When it comes to counting and measuring, as mentioned in the last paragraph, zero often does not come into play. Though technically measurements and counts should begin at zero – when nothing is there or has occurred yet – some traditional tools of measurement (such as rulers and stopwatches) do not show it. This absence causes some interesting dilemmas: for instance, there is much disagreement about whether the new century began on January 1, 2000 or January 1, 2001. The Library of Congress, the Royal Greenwich Observatory, and the National Institute of Standards and Technology all say that it was January 1, 2001, because the United States uses the Gregorian calendar (created by Pope Gregory in 1582), and on that calendar there is no year zero (the original one spans from 1 B.C.(E.) to 1 A.D. (C.E.). Other measurement tools, however, employ zero as a starting place. In races, the starting point is also called the “zero part” of the race. Zero is also the beginning of any measure of weight, like a scale when you’re weighing yourself.

Here is a list of curriculum resources for further investigation into the history of and concepts behind zero. I have noted which sites are most useful for teachers (T), students (S), or both (B).

Resources

O’Connor, J. and Robertson, E.F. (2000). “A History of Zero”. : St. Andrews Press. A relatively concise and engagingly written history of the discovery and ongoing use of zero as a number and as an indicator of place value. (T)http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Zero.html

Various Authors (Ongoing). Ask Dr. Math: The Math Forum. Philadelphia, PA: Drexel University. This website, from the Drexel University School of Education, has fantastic mathematics resources for students, teachers, and parents. There is a searchable archive as well as lesson plans and an interactive “Ask Dr. Math” section where visitors to the site can email or chat (live) about math with college and graduate school mathematics students and professors. (B)http://mathforum.org/dr.math/index.html

McQuillen, K. (1997, revised 2004). “A Brief History Of Zero”. Tokyo, : MediaTinker. A more extensive history of zero, with excellent diagrams, tables, and pictures. Strong upper elementary readers would be able to use it, but for the most part it would serve as a better resource for teachers (though students should definitely have the chance to see the reproductions of early symbols representing zero). (T) http://www.mediatinker.com/whirl/zero/zero.html

Author(s) and Year Unknown. “Zero”. Houghton Mifflin’s “Answers.com” gives a multitude of quick definitions and uses of zero, as well as an impressive list of translations and representations of zero (in every language from Dutch to Hebrew). (B) http://www.answers.com/zero

Arsham, H. (2002). “Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives”. Baltimore, MD: University of Baltimore Press. An absolutely fascinating article looking at zero from multiple perspectives. Parts could be reproduced for older elementary students’ use, but overall it serves the needs of teachers better. It would be a great resource for a really in-depth, interdisciplinary study of zero. (T)http://www.pantaneto.co.uk/issue5/arsham.htm

Author Unknown (2000-2005). “Zero Our Hero” from Illuminations. Reston, VA: National Council of Teachers of Mathematics. A set of seven early elementary lesson plans and activities for exploring and gaining experience with zero. It also has links to a couple of online activities for students. (B) http://illuminations.nctm.org/index_d.aspx?id=94#

There are also a number of other lessons relating to zero in Illuminations. Here are just a few:

– “Finding Sums to Six: The Additive Identity” http://illuminations.nctm.org/index_d.aspx?id=94#

– “Lost Buttons” http://illuminations.nctm.org/index_d.aspx?id=29

– “Fact Family Fun” http://illuminations.nctm.org/index_d.aspx?id=78

– “Exploring Adding With Sets” http://illuminations.nctm.org/index_d.aspx?id=54

– “Macaroni Math” http://illuminations.nctm.org/index_o.aspx?id=65

– “Looking Back and Moving Forward” http://illuminations.nctm.org/print_lesson.aspx?id=122

– “Multiplication Stories” http://illuminations.nctm.org/print_lesson.aspx?id=529

Wilson, P. (2001). “Zero: A Special Case” from Mathematics Teaching In The Middle School. Reston, VA: National Council of Teachers of Mathematics. (T)

Seife, C. (2000). Zero: The Biography of a Dangerous Idea. New York, NY: Penguin Group. Seife’s blend of anthropology, history, and math traces the journey of our most controversial number. Among other topics, it examines the connection between zero and infinity, and discusses in depth the various religious and academic denials and fears of zero as a number or a quantity. A better resource for teachers than students. (B)

French, V. and Collins, R. (2001). From Zero to Ten: The Story of Numbers. New York, NY: Oxford University Press. A kid-friendly book of the origins of numbers (including a separate chapter on zero, “Nothing matters now!” starting on page 14), with cute chapter headings like “Everything in the right place,” “Lucky numbers,” and “Using our bodies to measure.” (S)

Dorough, B. (1973) “My Hero, Zero” from Schoolhouse Rock. New York, NY: ABC. A very entertaining song from the creators of the cartoons Schoolhouse Rock, Grammar Rock, Science Rock, and Multiplication Rock. Created by an advertising executive who noticed that his son couldn’t remember his math facts but had no trouble remembering the words to every Beatles song ever written, “My Hero, Zero” (and the rest of the songs and animated shorts) explain math facts (such as the properties of zero) in a straightforward, easy-to-remember way. The cartoon and song can be downloaded in .wav format from the following website: http://www.school-house-rock.com/0.html. (S)

Lopresti, A. (2003). A Place For Zero: A Math Adventure. Watertown, MA: Charlesbridge. A cutesy (and sometimes overcomplicated) fictional personification of zero, who is alone in Digitaria because he can’t play Addemup (because he “has nothing to add”), and goes on a quest from Count Infinity’s workshop to King Multiplus’s kingdom. My second graders enjoyed this book a lot more than I did, but I am including it here because it seemed to work for a lot of the students. (S)

[1] National Council of Teachers of Mathematics “Principles and Standards for School Mathematics” http://standardstrial.nctm.org/document/chapter5/numb.htm

[2] National Council of Teachers of Mathematics “Principles and Standards for School Mathematics” http://standardstrial.nctm.org/document/chapter5/numb.htmhttp://standardstrial.nctm.org/document/chapter6/numb.htm

[3] Massachusetts State Frameworks and Learning Standards http://www.doe.mass.edu/frameworks/math/2000/final.pdf

[4] Ibid.

[5] Re-created from a concept found in McQuillen, K. (1997, revised 2004). “A Brief History Of Zero”. Tokyo, : MediaTinker. http://www.mediatinker.com/whirl/zero/zero.html

[6] O’Connor, J. and Robertson, E.F. (2000). “A History of Zero”. : St. Andrews Press. http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Zero.html

[7] Reprinted by McQuillen, K. (1997, revised 2004). “A Brief History Of Zero”. Tokyo, : MediaTinker. http://www.mediatinker.com/whirl/zero/zero.html

[8] Translation reprinted in O’Connor, J. and Robertson, E.F. (2000). “A History of Zero”. : St. Andrews Press. http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Zero.html

[9] Ibid.

[10] Ibid.

[11] Reprinted by McQuillen, K. (1997, revised 2004). “A Brief History Of Zero”. Tokyo, : MediaTinker. http://www.mediatinker.com/whirl/zero/zero.html