a) What is important about the concept of zero beyond its use as a place holder?

Question

a) What is important about the concept of zero beyond its use as a place holder?

b) What is the significance of using a particular symbol to represent an absence of quantity?

c) What role does zero play as a mathematical concept and how did it fit within the cultures as it was first understood?

d) What challenges were faced by cultures that did not have a zero in their number system and how were these challenges addressed?

e) What cultural influences come into play?

f) Was this new concept readily accepted and if not, what cultural influences might have prevented this acceptance?

Explanation / Answer

a)

importance of zero:-

It is the number around which the negative numbers to its left stretch into infinity and the positive numbers to the right do likewise. It is neither positive nor negative. For that reason, zero is a pivotal point on thermometers and is the origin point for bathroom scales and the coordinate axis.

Zero is also important when you think of sets. An empty or null set is one which has no items in it.

Zero is so important that each of the mathematical operations has special rules called properties governing its use with other whole numbers.

b)

Judging from the treatment accorded to the concept of zero, we do practice a variety of avoidance mechanisms rather than confront the imagery associated with this seemingly difficult concept.

In reciting one's telephone number, social security number, postal zip code or post office box, room number, street number or any of a variety of other numeric nominals, we carefully avoid pronouncing the digit "zero" and instead substitute "oh." One may say "it is caused by our desire to communicate quickly, if we can say the same thing in one syllable, why not?" What about number seven, should we find a substitute for this too?

In some parts of the world, the phrasing "naught" and "aught" are used but it is quite uncommon to hear "zero." All the other digits are correctly enunciated with this one curious exception.

Is the presence of nothing (reflecting non-existence) different from the absence of something (reflecting non-availability) or the absence of anything (reflecting non-existence)? Zero is a symbol for "not there" which is different from "nothing" "Not there" reflects that the number or item(s) exists but they are not just available. "Nothing" reflects nonexistence.

Zero not only has the quality of being nothing, it is also a noun, verb, adverb, and an adjective as in "zero possibility". "We zeroed in on the cause," means we had isolated all the possibilities, and have discovered the one remaining. In this use as a verb, zero equals one. However, "The result was a big, fat, zero," uses the noun to express the idea of results of "nothing". Here, zero has the quality of not being there. Zero as an action appears in the Conservative Laws of physics.

c)

There are at least five aspects to being able to understand place-value, only two or three of which are often taught or stressed. The other two or three aspects are ignored, and yet one of them is crucial for children's (or anyone's) understanding of place-value, and one is important for complete understanding, though not for merely useful understanding. I will first just name and briefly describe these aspects all at once:-

1) Learning number names (and their serial order) and using numbers to count quantities, developing familiarity and facility with numbers, practicing with numbers --including, when appropriate not only saying numbers but writing and reading them, not in terms of rules involving place-value, etc., but in terms of just being shown how to write and read individual numbers (with comments, when appropriate, that point out things like "ten, eleven, twelve, and all the teens have a '1' in front of them; all the twenty-numbers have a '2' in front of them" etc.) without reasons about why that is,

2) "simple" addition and subtraction,

3) developing familiarity through practice with groupings, and counting physical quantities by groups (not just saying the "multiples" of groups -- e.g., counting things by fives, not just being able to recite "five, ten, fifteen,..."), and, when appropriate, being able to read and write group numbers --not by place-value concepts, but simply by having learned how to write numbers before. Practice with grouping and counting by groups should, of course, include groupings by ten's,

4) representation (of groupings)

5) specifics about representations in terms of columns.

Aspects (1), (2), and (3) require demonstration and "drill" or repetitive practice. Aspects (4) and (5) involve understanding and reason with enough demonstration and practice to assimilate it and be able to remember the overall logic of it with some reflection, rather than the specific logical steps.

d)

The introduction of zero into the decimal system in 13th century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. Without the notion of zero, the descriptive and prescriptive modeling processes in commerce, astronomy, physics, chemistry, and industry would have been unthinkable. The lack of such a symbol is one of the serious drawbacks in the Roman numeral system. In addition, the Roman numeral system is difficult to use in any arithmetic operations, such as multiplication. The purpose of this article is to raise students, teachers and the public awareness of issues in working with zero by providing the foundation of zero form four different perspectives. Imprecise mathematical thinking is by no means unknown; however, we need to think more clearly if we are to keep out of confusions.

Our discomfort with the concepts of zero (and infinite) is reflected in such humor as 2 plus 0 still equals 2, even for large values, and popular retorts of similar tone. A like uneasiness occurs in confronting infinity, whose proper use first rests on a careful definition of what is finite. Are we mortals hesitant to admit to our finite nature? Such lighthearted commentary reflects an underlying awkwardness in the manipulation of mathematical expressions where the notions of zero and infinity present themselves. A common fallacy is that, any number divided by zero is infinity. It is not simply a problem of ignorance by young novices who have often been mangled. The same errors are commonly committed by seasoned practitioners, yea, and even educators! These errors frequently can be found as well in prestigious texts published by mainstream publishers.

e)

cultural influences:-

a) It may be considered frivolous hyperbole to suggest that the demise of the Roman Empire was due to the absence of zero in its number system, but one can only ponder the fate of our civilization given the difficulty our culture seems to have with the presence of zero in our number system.

b) The notion of zero brings another wearying and yet intriguing questions: Is our current century the 20th century or the 21st century? According to the Holy Scriptures (see, Matthew chapter 2), King Herod was alive when Jesus was born, and Herod died in 4 BC. Does that mean the millennium actually started in 1996?

c) Ordinal numbers, which the Gregorian calendar uses, indicate sequence. Thus "A.D. 1" (or the first year A.D.) refers to the year that begins at the zero point and ends one year later. Think of a carpenter's ruler, if you will; the first inch is the interval between the edge and the one-inch mark. Thus, e.g., the millennium ended with the passing of the two-thousandth year, not with its inception. Cardinal numbers, which astronomers use in their calculations, indicate quantity. Zero is a cardinal number and indicates a value; it does not name an interval. Thus "zero" indicates the division between B.C. and A.D., not the interval of the first year before or after this point. Continuing with our example, put two rulers end to end: although there is a zero point, there is no "zero'th" inch.

d) The main confusion is between the notions of "time window length" and a "point in time". There is an interval between 0 and 1. Considering whether this century is 2000 or 2001, depends on whether you look at a number as a points on time or a time interval. Years are intervals; numbers are points. Therefore, it is always a mistake to treat years as points. For example, consider the old arithmetic question: John was born in 1985 and Jane in 1986.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.