Thomas Okay. Burch [1] from the College of Victoria [Canada] writes about the advantages of utilizing engineering notation in demography.
Introduction
Many scientific disciplines routinely use scientific notation when coping with very giant or very small numbers. Portions are expressed as a quantity between 1 and 10 [the coefficient] instances an influence of 10. For instance, the velocity of sunshine in a vacuum is estimated at 299,792,498 metres per second. In scientific notation, this may be roughly 3.0 x 108. This abbreviated or shorthand type of the unique quantity is simpler to recollect and to say or write. It additionally makes it simpler to maintain observe of zeros or decimal locations in calculations.
Though to not the identical extent as physics or astronomy, demography additionally entails comparatively giant and small numbers [for example, world population [7,800,000,000], or the likelihood of dying of a 10-year-old feminine in a low-mortality inhabitants [0.00068], however using scientific notation within the demography is uncommon.[2]
Of more moderen origin and fewer extensively used is engineering notation. That is analogous to scientific notation however permits coefficients between 1 and 1,000 and limits exponents to multiples of three. The exponents correspond to acquainted metrics similar to hundreds, tens of millions, billions, and so forth. for giant numbers, and thousandths, millionths, billionths, and so forth. for small numbers. In engineering notation, for instance, the velocity of sunshine [299,792,498] can be 229.8 x 106. The exponent 6 signifies tens of millions; coefficient 229.8 provides the quantity.
A better take a look at these two methods of abbreviating numbers suggests engineering notation stands out as the extra great tool for demography.
Scientific Notation
A quantity in scientific notation consists of two components forming a product. The primary is a quantity between 1 and 10, known as the coefficient or issue. The second is an integral energy of 10, typically known as the exponential time period. For instance, 256,000 turns into 2.56 x 105 or 2.56 x 10,000. No matter enter, the coefficient is compelled to be between 1 and 10. Thus, the quantity 2,560,000 [10 times 256,000] is abbreviated as 2.56 x 106. The coefficient stays the identical; the exponent is elevated.[3] The coefficient could also be carried out to as many decimal locations as are justified by the accuracy of the information. However usually, coefficients in scientific notation are restricted to 2 or three important digits.[4]
Engineering Notation
Engineering notation differs from scientific notation in two respects. First, the coefficient will be any quantity between 1 and 1,000. Second, the exponent have to be a a number of of three. These exponents are related to acquainted portions: 3 signifies hundreds, 6 tens of millions, 9 billions, and so forth., or on the damaging facet, -3 signifies thousandths, -6 millionths, and so forth. If the coefficient is lower than 1 or larger than 1,000, the exponent is lowered or raised to the following a number of of three.[5]
Evaluating Scientific and Engineering Notation
It’s instructive to look at the 2 sorts of notation throughout a variety of enormous and small numbers which may come up in demographic evaluation [see Fig. 1].[6]
The 2 notations are equally simple to recollect, write or communicate, and they’re equally efficient in retaining observe of zeros and decimal factors. One benefit of scientific notation is that it makes it simple to check two numbers. As a result of the coefficient have to be between 1 and 10, numbers with larger exponents are bigger than these with smaller exponents. And ‘…subtraction of exponents provides an estimate of the variety of orders of magnitude separating the numbers’ [Wikipedia, s.v. Scientific notation]. Typically, engineering notation is simpler to interpret in substantive phrases. The exponent signifies the metric [thousands, millions, billions…]; the coefficient signifies their quantity in 1, 2 or 3 digits. Numbers in engineering kind instantly match their verbal descriptions; numbers in scientific notation don’t.
In scientific notation, for instance, the present inhabitants of Canada is 3.82 x 107, an expression not instantly recognizable as 38.2 million. The end result have to be modified, mentally or in writing. The engineering notation is 38.2 x 106. Th exponent 6 signifies tens of millions and 38.2 provides their quantity. Engineering notation is extra clear.
Equally, the present beginning price for Canada, 9.9 per 1,000 in typical phrases, will be expressed as 9.9 x 10-3. The time period 10-3 in impact means ‘per thousand.’ There’s a sense during which this normal demographic conference for crude charges is a step in the direction of engineering notation. However the conventional phrase ‘9.9 births per 1,000 inhabitants’ can’t be utilized in calculation, whereas 9.9 x 10-3 can.
Calculation with Numbers in Scientific and Engineering Notation
The widespread follow in demography and associated fields is and has been to calculate with numbers in decimal notation. Massive numbers are written with commas or areas inserted to assist in retaining observe of digits and zeros. The usage of commas or areas in small decimal fractions is much less widespread, however not unknown. Previous follow is much less helpful in an age of digital calculators and computer systems, lots of which don’t help using commas or areas and favor using scientific and engineering notation in each enter and output.[7]
Calculation in scientific and engineering notation follows the final guidelines for exponents. Addition and subtraction can happen solely between numbers with the identical exponents. For instance, 2 x 103 [2,000] can’t be added to three x 104 [30,000] with out altering the coefficient and exponent in one of many components. Re-writing the primary issue by transferring the decimal one place to the left and elevating the exponent by 1 provides 0.2 x 104. The sum is now 3.2 x 104 [32,000].
In multiplication coefficients are added. In division the exponent within the denominator is subtracted from that within the numerator. Each processes contain three steps: a] calculation of the product or quotient of the coefficients; b] calculation of the product or quotient of the exponential phrases; c] calculation of the product of the 2 outcomes. On some calculators and in most calculation software program these steps are finished concurrently.
Fig. 2 reveals the three-step process for division and multiplication, with estimates for world inhabitants, births, and the crude beginning price round 2020. Clearly, all of the digits within the enter information are usually not correct. Precision in most calculations has been arbitrarily restricted to a few or 4 digits.
The primary part of Fig. 2 illustrates division, calculating the CBR because the ratio of births to inhabitants with the three-step process and with scientific notation. The results of the three-step process is 0.1796 x 10-1, 0.01796 in decimal notation, or 17.96 per 1,000.
The pc calculation in scientific notation doesn’t observe the usual guidelines for exponents: 8 [numerator] minus 9 [denominator] equals -1; however the reply reveals 10-2. The quotient of the coefficients is 0.1796. This system forces it into scientific notation, with a coefficient of 1.796; the exponent have to be lowered to -2. The direct interpretation of the end result
With engineering notation births will be interpreted readily as 140 million, and inhabitants as 7.8 billion. The calculation of the CBR follows the same old guidelines: the ratio of the coefficients is 17.96; that of the exponential phrases is 10-3 [6 – 9 = -3]. The reply interprets instantly as 17.96 births per 1,000 inhabitants. General, calculation with engineering notation is extra orderly and extra clear.
Comparable outcomes get hold of with multiplication, for instance, estimating the variety of births given the CBR and whole inhabitants. The three-step process and engineering notation each return 140 x 106 or 140 million births. The end in scientific notation is 1.4 x 108. The coefficient is compelled to a quantity between 1 and 10, and the exponent raised to eight.
Abstract and Conclusions
The arrival of digital calculators and computer systems [with spreadsheets and mathematics software] has tremendously facilitated demographic calculation. Previously laborious computations similar to the development of a life desk are actually comparatively fast and simple with using a normal spreadsheet similar to Excel.
However some calculations can nonetheless be difficult – these involving very giant or very small numbers, or numbers given in several orders of magnitude [e.g., population in millions and a rate per thousand]. It may be tough to maintain observe of zeros and/or decimal factors, particularly for the coed or novice. A bonus of scientific notation is that it makes this job simpler.
For demography, nonetheless, scientific notation has some disadvantages. Most vital, the limitation of the coefficient to values between 1 and 10 yields numbers that don’t translate on to acquainted demographic metrics – hundreds, tens of millions, billions, per thousand, per million, and so forth. When the necessity for abbreviated numbers arises, demography is best served by engineering notation, the place the coefficient will be one, two or three digits, and the exponent corresponds to those self same metrics. Engineering notation is also extra according to the fundamental guidelines for multiplication and division involving exponents. It’s extra clear and extra orderly. It might be a helpful addition to the demographer’s toolkit.
[1] See Appendix for biographical sketch.
[2] There isn’t any point out of it within the main demography texts.
[3] On some calculators, this is likely to be displayed as 2.56E6. Word that E isn’t associated to the fixed e.
[4] See Wikipedia, s.v. ‘Scientific Notation.’
[5] See Wikipedia, s.v. ’Engineering Notation.’
[6] Fig. 1 is a Mathcad Prime 6 worksheet pasted into Phrase. It’s not a stay mathematical worksheet.
[7] This refers to built-in procedures. Usually, extensions or workarounds are potential. In R. for instance, there’s a program that facilitates using engineering notation, together with in tables. https://cran.r-project.org/net/packages/docxtools/vignettes/numbers-in-engineering-format.html
Appendix – Biographical Sketch:
Thomas Okay. Burch is Adjunct Professor, Dept. of Sociology, College of Victoria [Canada] and a Regional Affiliate, Middle for Research in Demography and Ecology, College of Washington. He’s Professor Emeritus, Western College, London, Ont., the place he served from 1975 to 2000, working to develop a Ph.D. program in social demography. Earlier than becoming a member of the school at Western, he taught at Marquette Univ., Milwaukee, Wisc. And Georgetown College, Washington, D.C. He has been a visiting professor on the Univ. of California, Berkeley and on the Univ. of Rome, and a visitor researcher on the Max Planck Institute for Demographic Analysis, Rostock, Germany. From 1970 to 1975 he was Affiliate Director, Demographic Division, The Inhabitants Council, New York, NY.
His essential areas of analysis and publication are fertility, marriage and divorce, and family and household. Late in his profession he targeted on a methodological evaluation of demography as a science, with specific consideration to principle and fashions. A latest abstract of this work is Mannequin-Primarily based Demography: Essays on Integrating Information, Approach and Principle, Max Planck Institute for Demographic Analysis and Springer Open, 1918 [open access]. In a first-year college course in natural chemistry, the creator used a slide-rule for all calculations. As a graduate pupil within the late Nineteen Fifties, he used mechanical calculators that will add, subtract, multiply and divide – solely – with no tapes or printouts. Calculations involving logarithms, exponentials or roots required consulting giant reference books of tables. The pc revolutionized demographic calculation.
