Chapter 9. Talking about
Numbers or Quantities

Volume 2, Three Skills for Algebra

We can identify and speak about numbers and quantities without doing any arithmetic. Words from everyday speech can be used to talk about them. Quantities are also called amounts. Words have been missing to introduce and describe the algebraic way of writing and reasoning.

The emphasize on words here before or beside symbols introduces a new dimension or topic in understanding and explaining  mathematics in general, the deliberate and hopefully earlier description of numbers and amounts apart from arithmetic and algebra.

Words have been missing to introduce and describe the algebraic way of writing and reasoning. The following sections of this chapter offer words to begin learning or teaching the algebraic way of writing and reasoning. Enter one section. Then use the next, previous links in these pages to move between them..

1.  Identifying Numbers and Quantities

We first identify some numbers and quantities. After this, perhaps, we can speak about them, or describe them, all without doing any arithmetic. There is more to mathematics than just doing arithmetic.

Here are a few not-too-serious examples of numbers and quantities. Height is a quantity. A building has a height. So has an elephant. The elephant also has a weight and a width or a girth. A rectangle has a length, a width and an area. A closed box has a width, a length, a height and a volume. The people in a room or in a town can be counted. This gives us a number. The difference between a number and a quantity will be explained later. More examples of numbers and quantities follow.

1. The amount of money in a bank account (measured in dollars, pounds, yen, etc.)
2. the depth of a swimming pool (measured in inches, feet, yards, centimeters, meters, etc, whereever these units are in used).
3. The height of an airplane (measured in feet or meters).
4. The radius of a wheel (measured in whatever units you like).
5. The number of goats in a field (a count - no units).
6. The number of feet in your height.
7. The number of meters in your height - not the same as the number of feet!
8. The amount of money you have (in your local currency).
9. The speed of a car now (measured in miles per hour, feet per second, meters per second, or kilometers per hour, etc.)
10. The radius, area and perimeter (distance around) of a circle (measured in feet, inches, centimeters, kilometers, etc.)
11. The height, width and length and volume of a box (measured in various units).
12. The rate of interest your savings get - compounded or simple, measured in percent or given by a decimal number, etc.
13. The number of days in this month - whatever month it is, a whole number depending on the month and, in the case of February, depending on the year as well.
14. The distance between you and your home (measured in miles, kilometers, etc.)
15. The time required for a journey (measured in seconds, minutes, hours, days, weeks, etc.)

This list could continue. We have identified several numbers and quantities. We can talk and think about these numbers and quantities although we have not seen and we have not measured them.

2.  Using Everyday Words

Our next aim is to show how everyday words should be used in mathematics to describe numbers and quantities - their use here is close to their everyday meanings. For example, we can say if a number or quantity is known or not, changing or not, constant or not, increasing, decreasing, shrinking, growing, confidential or embarrassing, top-secret or simply forgotten. Everyday words give the descriptive vocabulary of mathematics. Describing and talking about quantities and numbers is a part of mathematics after arithmetic. More examples follow.

2.1  Airplanes or Jets

2.2  People

We can also speak about the number of people in a room. When nobody enters or leaves, this number remains constant. When somebody enters or leaves, this number varies. This number or count is usually a whole number or zero. When someone is just leaving and partly in and partly out of this room, we cannot count or we have to allow fractions.

When we speak about the number of people in a room do we mean completely in, do we include fractions, or do we just say the count cannot be done at those moments when someone is partly in or out, moving or not? This number or count needs to be clearly defined. Words are needed to say precisely how it is computed, otherwise ambiguity results.

2.3  Height

When a building is being constructed, its height is increasing. The construction and the increase in height of the building may take place over one or two years. While the building is used, say seventy years, its height may be constant - unchanging. At the end of the building's useful life, the building is left to fall down or it is demolished - torn down. Here over a long or short time, the height decreases.

The height of the building varies. This height is therefore a variable during the construction and the demolition (collapse or falling down) of the building. The height is usually a constant, unchanging and invariable quantity during the seventy or so years that the building is used.

The height of the building may or may not be known to us during the lifetime of the building. Yet we can still refer to the height of the building, and to its other dimensions, even if we have not measured these quantities and even if they are unknown to some or all of us.

Here are some more questions, just for fun. What do we mean by the height of the building? Before the building is built, can we talk about its height? Can the height be taken to be zero? When the building is being built, is the height of the building equal to the height of its walls as they are being put up? If the building has a basement or a foundation, do we say the height of the building is negative or is it undefined while the basement is being dug, or the foundations being built? When the building is being demolished, does it have a height? What is it?

What do we mean by height? Better yet, we can speak of the height of a building whenever we can say what it represents (means) and/or how we might measure it. This permits us to speak of the current height, the planned or intended height, the past height, the future height. Is the height of a demolished building zero, or undefined? Is the planned height of a building equal to its actual height before construction, during construction, during its use or during demolition? A definition or identification of the height we want to speak about, is needed.

3  Mathematical Usage of Words

The above examples show how everyday words are used to describe numbers and quantities. Our next task is to say further or more precisely how the words variable, constant, known and unknown are used to both describe and refer to numbers and quantities.

See the postscript: What is a Variable

A. Variables Versus Constants

To say that a number or quantity is variable means that the number or quantity may vary or change. To say that a number or quantity is constant means that its value remains unchanged. For example:

1.  

2. The Greek letter π= 3.14159... (approximately) stands for or denotes a constant - a value or number which will never change.
3. The time of day is always changing. So time is varying. It is an example of a number or quantity which is always increasing and therefore variable. When you ask what time it is, you will get an approximate answer.

To complicate matters further, numbers and quantities may change in one period and not in another. The height of house increases slowly as it built, remains constant while it used, and decreases rapidly if it torn down. So this height may be variable in some situations and constant in others. In everyday life and in mathematics, when a number or quantity is called a constant, we expect its value not to change in the situation at hand. Similarly, when a number or quantity is called a variable, we should expect or suspect that its value may change.

More examples: Your height is a variable or it was a variable while you were growing. The speed of a car or a bicycle is an example of a variable (a variable number or quantity that is). The speed of a car can be almost constant. The zero speed of a stationary car or a parked car is constant - in one reference system at least. Note that a number or quantity can be variable in one situation, and constant in another. We can further talk about a previously constant or a previously variable number or quantity.

In summary, the terms constant and variable can be used to talk about and describe numbers and quantities. A constant is a number or quantity whose value is expected not to change - whose value should not change. A variable is a number or quantity whose value does or might change. The use of these terms is flexible and context dependent. What is constant in one situation may be varying or changing in another.⁵

⁵In some algebra texts and in some dictionaries, the term variable means or refers to the letters that appear in formulas. That use of the term variable departs from the use and meaning given above. In my view, the mathematical usage of everyday words should be in the first instance linked and extracted from their ordinary usage. Where the mathematical usage has departed from the everyday usage, we need to ask if that departure is necessary, and whether or not the departure should be corrected. Documenting reasons or possibly causes for such departures could be material for a thesis in linguistics.

B.  Known Versus Unknown

Numbers and quantities can be known or unknown. You may know your own height, age and weight, but I don't know your personal measurements. To you these quantities are known. To me they are unknown. Whether they are known or not depends on the company you keep - that is to whom you speak. When you see the instruction find the unknown, you should ask the question: unknown to whom? Note further in solving an equation, the solution of the equation goes from being unknown to being known.

C. What is a  Parameter?

For the sake of variety in our speech, numbers and quantities are also called parameters. A parameter is another name for a number or a quantity. When we say a number or quantity is a parameter, we have no immediate expectation that the number or quantity in question will be constant nor that it will be variable. The term parameter gives a vague expectation somewhere between constant and variable. We can talk about numbers and quantities in precise and imprecise ways.

4  Approximate Knowledge

Numbers and quantities are known, given, measured or estimated with varying precision. For instance, the cost of a hot dog could be 2.25 dollars. This cost is given exactly. In contrast, the height of a man might be between 5[1/2] and 6 feet and the weight of a truck could be between one and ten tons. In these two cases, the quantities in question are sandwiched or bracketed between two extreme values: the least and greatest possible. (The term sandwiched is preferred. It is more graphic.) The distance between the bracketing values measures the uncertainty in our knowledge.

⁷NOTE FOR ADVANCED STUDENTS: More precisely, if x is a number whose value is known to be between two positive number a and b with a £ b, then the mean value c = [(a+b)/2] gives an approximation to x. The absolute error in this approximation is £ [1/2]|b-a|. The percentage error in this approximation is £ 100·[1/2][(|b-a|)/(a)]%. The relative error in this approximation is £ [1/2][(|b-a|)/(a)]. To say that the percentage error is at most 1% indicates a better approximation than a percentage error of at most 5% or even 100%. In the above examples, note for instance the following: The height of the man is known within 100[(0.25ft)/(5.5ft)] = 4.55% £ 5%, a small (?) uncertainty. The weight of the truck is known within 100[(4.5tons)/(1ton)] = 450%. The uncertainty in the latter is large.The symbol £ is shorthand for the expression less than or equal to.

Knowledge of numbers and quantities may be exact or approximate. But we can still speak about them. We can also use approximate values in calculations and then hope the resulting error is not too large. Estimating errors in calculations is a useful topic which cannot be fully explored here. Error estimation is limited by the observation that perfect knowledge of the error in a computation would provide a means for removing the error. So error estimates must remain imperfect.

⁸Significant Digits etc: When you say that the height of a building is 10.47 meters (approximately) without giving any further information, the uncertainty in the last digit 7 should be £ [1/2]. When a single decimal is used to approximate a number or quantity, the digits in it are said to be significant when and only when the uncertainty in the last digit written is £ [1/2] of a unit. Digits which are uncertain by more than [1/2] should not be written when we report the result of a measurement or calculation.

Exception: When a single quantity x is bracketed between two others, say a and b, their mean value c = [(a+b)/2] provides an approximation to x with an error of at most d = [(|b-a|)/2]. In this case we may write x = c±d and keep some digits in the decimal expansion of c with an uncertainty in them of more than one half unit. Writing x = (10.472±0.003) meters for example provides more information about x than the single estimate x = 10.47 meters.

In some situations, the location of the last digit with an uncertainty of less than [1/2] of a unit may be unknown and this convention may be difficult to follow. Errors in long calculations may be minimized if rounding-off is postponed as long as possible, for instance done at the end of all calculations and not for intermediate results.

Another Example: In crossing a toll bridge with one rate for trucks weighing under 10 tons and with a higher rate for trucks over 10 tons, the knowledge that the truck is between one and ten tons means that the lower rate is used. But in crossing a bridge with a higher toll rate for trucks over five tons, the knowledge that the truck is between one and ten tons is not accurate enough. The truck has to be reweighed.

5.  Numbers Versus Quantities

When you ask how tall I am, you may get the answer: 5 feet and 10 inches or 1.75 meters. The answer in either of its forms involves both numbers and units. A number times a unit of measurement gives you a quantity. Quantities can be added together: 5 feet plus 10 inches is 5 and 5 sixths feet.

To further understand the difference between numbers and quantities, you may ask how many pennies (or cents) I have in my pocket. The answer could be the number 10. For the same pocket, if you asked how much money I had in it, the answer would be the quantity 10 cents or even 0.10 dollars, a tenth of a dollar.

Numbers are given by counts - whole numbers, proper and improper fractions, decimal numbers. Quantities are given by a count (a whole number or fraction) times a unit of measurement. Any object that can be counted can serve as a unit of measurement. Examples of units of measurement are: meter, foot, $ or dollar, square foot, square meter, second, hour, meters per second, kilometers per hour, dollars per hour, miles per hour and so on.

Numbers include no units. You get a number when you ask how many units there are, and you have specified the unit. You get a quantity when you ask how much there is. Saying a length is given by the number 5 is meaningless, if no units of measurement are given. Saying a length is 5 raises the question 5 what?

The number 5 may give the number of units of length in a distance. Writing this number by itself does not say what the unit of length might be. Some information, the unit, is missing. So I repeat, in answering questions demanding how much, we need to give a unit of measurement as well as a number. People should not have to guess your unit of measurement when you speak. A length may be given by 5 miles (or 8 kilometers). Of course, if we are asked how many miles (or kilometers) there are in the length concerned, the number 5 (or 8) is expected because the unit was specified. When you are asked how many people there are in a room, you may respond with a pure number like 7 or 10. The unit of measurement can be worded or written as person or persons.

In measurement and counting, a single unit of measurement, a fraction of one or several units, may appear. For instance, a length of time may involve 1 hour or 12.5 hours. Notice the addition of the letter s to the unit hour here when fractions or more than one unit appears. In mathematics, we choose to ignore the difference in spelling between the singular and the plural. If we insisted on using the singular form, we would have to write 12.5 hours = 12.5 ×1 hour. The latter gives the exact meaning of 12.5 hours. In writing units in calculations, we may and will change their spelling (or abbreviations) according to the rules of grammar. The plural and singular forms of each unit are declared to be equal or interchangeable. Each is allowed to replace the other. Which one sounds the most appropriate will be written in our formulas and calculations.

5.1  Changing Units

In reporting quantities or measurements, the choice of units affects the number of units. Changing the unit of measurement will change the number. For example, the amount of time 2 hours is the same as the amount of time 120 minutes, but the numbers 2 and 120 are different. We can say we have two units of time when the time is measured in hours. We can say we have 120 units of time when time is measured in minutes. But we cannot say the amount of time is 2 or 120 without saying or somewhere saying what unit of measurement is used. The following example shows how to change the unit of measurement.

To express the quantity 10.5 hours in terms of minutes, we replace 1 hour by

60 minutes = 60 × 1 minute

This gives

10.5  hours = 10.5 ×1  hour = 10.5 ×( 60×1  minute ) = 10.5 ×60  minutes = 630  minutes.

6  Review and References

We have seen the mathematical use of several words including: variable, constant, changing, unchanging, non-varying, unknown, known, etc. Speaking about numbers and quantities without doing any arithmetic is part of mathematics. It is also part of any subject involving calculations.

The following books may help you review or practice your mathematical skills. ( A visit to a local book store showed me there are many, more recent books. Still more appear everyday.)

1.  

2. Arithmetic Made Simple by A. P. Sperling & S. D. Levinson, 1988, Doubleday 1960 & 1988, 666 Fifth Avenue, New York, New York 10103. ISBN 0-385-23938-6.
3. Arithmetic Refresher for Practical People by A. A. Klaf, 1964, Dover, New York, ISBN 0-486-21241-6
4. Mathematics for Practical Use by Kaj L. Nielsen, Barnes & Noble 1962.
5. Short-Cut Math by Gerard W. Kelly, 1984, Sterling Publishing Co, ISBN 0-486-24611-6.
6. Helping your child with Maths by R. S. Harrison, 1982, Harrap Limited, ISBN 0 245-53802-X

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