*Thomas K. Burch* [1] *from the University of Victoria [Canada] writes about the benefits of using engineering notation in demography.*

**Introduction**

Many scientific disciplines routinely use scientific notation when dealing with very large or very small numbers. Quantities are expressed as a number between 1 and 10 [the coefficient] times a power of 10. For example, the speed of light in a vacuum is estimated at 299,792,498 metres per second. In scientific notation, this would be approximately 3.0 x 10^{8}. This abbreviated or shorthand form of the original number is easier to remember and to say or write. It also makes it easier to keep track of zeros or decimal places in calculations.

Although not to the same extent as physics or astronomy, demography also involves relatively large and small numbers [for example, world population [7,800,000,000], or the probability of death of a 10-year-old female in a low-mortality population [0.00068], but the use of scientific notation in the demography is rare.[2]

Of more recent origin and less widely used is engineering notation*.* This is analogous to scientific notation but allows coefficients between 1 and 1,000 and limits exponents to multiples of 3. The exponents correspond to familiar metrics such as thousands, millions, billions, etc. for large numbers, and thousandths, millionths, billionths, etc. for small numbers. In engineering notation, for example, the speed of light [299,792,498] would be 229.8 x 10^{6}. The exponent 6 indicates millions; coefficient 229.8 gives the number.

A closer look at these two ways of abbreviating numbers suggests engineering notation may be the more useful tool for demography.

**Scientific Notation **

A number in scientific notation consists of two parts forming a product. The first is a number between 1 and 10, called the coefficient or factor. The second is an integral power of 10, sometimes referred to as the exponential term. For example, 256,000 becomes 2.56 x 10^{5 }or 2.56 x 10,000. Regardless of input, the coefficient is forced to be between 1 and 10. Thus, the number 2,560,000 [10 times 256,000] is abbreviated as 2.56 x 10^{6}. The coefficient remains the same; the exponent is increased.[3] The coefficient may be carried out to as many decimal places as are justified by the accuracy of the data. But generally, coefficients in scientific notation are limited to two or three significant digits.[4]

**Engineering Notation**

Engineering notation differs from scientific notation in two respects. First, the coefficient can be any number between 1 and 1,000. Second, the exponent must be a multiple of 3. These exponents are associated with familiar quantities: 3 indicates thousands, 6 millions, 9 billions, etc., or on the negative side, -3 indicates thousandths, -6 millionths, etc. If the coefficient is less than 1 or greater than 1,000, the exponent is lowered or raised to the next multiple of 3.[5]

**Comparing Scientific and Engineering Notation**

It is instructive to examine the two kinds of notation across a range of large and small numbers that might arise in demographic analysis [see Fig. 1].[6]

The two notations are equally easy to remember, write or speak, and they are equally effective in keeping track of zeros and decimal points. One advantage of scientific notation is that it makes it easy to compare two numbers. Because the coefficient *must be* between 1 and 10, numbers with bigger exponents are larger than those with smaller exponents. And ‘…subtraction of exponents gives an estimate of the number of orders of magnitude separating the numbers’ [*Wikipedia*, s.v. Scientific notation]. Generally, engineering notation is easier to interpret in substantive terms. The exponent indicates the metric [thousands, millions, billions…]; the coefficient indicates their number in 1, 2 or 3 digits. Numbers in engineering form directly match their verbal descriptions; numbers in scientific notation do not.

In scientific notation, for example, the current population of Canada is 3.82 x 10** ^{7}**, an expression not immediately recognizable as 38.2 million. The result must be modified, mentally or in writing. The engineering notation is 38.2 x 10

^{6}. Th exponent 6 indicates millions and 38.2 gives their number. Engineering notation is more transparent.

Similarly, the current birth rate for Canada, 9.9 per 1,000 in conventional terms, can be expressed as 9.9 x 10^{-3}. The term 10^{-3} in effect means ‘per thousand.’ There is a sense in which this standard demographic convention for crude rates is a step towards engineering notation. But the traditional phrase ‘9.9 births per 1,000 population’ cannot be used in calculation, whereas 9.9 x 10^{-3} can.

**Calculation with Numbers in Scientific and Engineering Notation**

The common practice in demography and related fields is and has been to calculate with numbers in decimal notation. Large numbers are written with commas or spaces inserted to aid in keeping track of digits and zeros. The use of commas or spaces in small decimal fractions is less common, but not unknown. Past practice is less useful in an age of electronic calculators and computers, many of which do not support the use of commas or spaces and favor the use of scientific and engineering notation in both input and output.[7]

Calculation in scientific and engineering notation follows the general rules for exponents*.* Addition and subtraction can occur only between numbers with the same exponents. For example, 2 x 10^{3} [2,000] cannot be added to 3 x 10^{4} [30,000] without changing the coefficient and exponent in one of the factors. Re-writing the first factor by moving the decimal one place to the left and raising the exponent by 1 gives 0.2 x 10^{4}. The sum is now 3.2 x 10^{4} [32,000].

In multiplication coefficients are added. In division the exponent in the denominator is subtracted from that in the numerator. Both processes involve three steps: a] calculation of the product or quotient of the coefficients; b] calculation of the product or quotient of the exponential terms; c] calculation of the product of the two results. On some calculators and in most calculation software these steps are done simultaneously.

Fig. 2 shows the three-step procedure for division and multiplication, with estimates for world population, births, and the crude birth rate around 2020. Clearly, all the digits in the input data are not accurate. Precision in most calculations has been arbitrarily limited to three or four digits.

The first section of Fig. 2 illustrates division, calculating the CBR as the ratio of births to population with the three-step procedure and with scientific notation. The result of the three-step procedure is 0.1796 x 10^{-1}, 0.01796 in decimal notation, or 17.96 per 1,000.

The computer calculation in scientific notation does not follow the standard rules for exponents: 8 [numerator] minus 9 [denominator] equals -1; but the answer shows 10^{-2}. The quotient of the coefficients is 0.1796. The program forces it into scientific notation, with a coefficient of 1.796; the exponent must be lowered to -2. The direct interpretation of the result

With engineering notation births can be interpreted readily as 140 million, and population as 7.8 billion. The calculation of the CBR follows the usual rules: the ratio of the coefficients is 17.96; that of the exponential terms is 10^{-3} [6 – 9 = -3]. The answer translates directly as 17.96 births per 1,000 population. Overall, calculation with engineering notation is more orderly and more transparent.

Similar results obtain with multiplication, for example, estimating the number of births given the CBR and total population. The three-step procedure and engineering notation both return 140 x 10^{6} or 140 million births. The result in scientific notation is 1.4 x 10^{8}. The coefficient is forced to a number between 1 and 10, and the exponent raised to 8.

**Summary and Conclusions**

The advent of electronic calculators and computers [with spreadsheets and mathematics software] has greatly facilitated demographic calculation. Formerly laborious computations such as the construction of a life table are now relatively quick and easy with the use of a standard spreadsheet such as Excel.

But some calculations can still be challenging – those involving very large or very small numbers, or numbers given in different orders of magnitude [e.g., population in millions and a rate per thousand]. It can be difficult to keep track of zeros and/or decimal points, especially for the student or novice. An advantage of scientific notation is that it makes this task easier.

For demography, however, scientific notation has some disadvantages. Most important, the limitation of the coefficient to values between 1 and 10 yields numbers that do not translate directly to familiar demographic metrics – thousands, millions, billions, per thousand, per million, etc. When the need for abbreviated numbers arises, demography is better served by engineering notation, where the coefficient can be one, two or three digits, and the exponent corresponds to those same metrics. Engineering notation also is more consistent with the basic rules for multiplication and division involving exponents. It is more transparent and more orderly. It could be a useful addition to the demographer’s toolkit.

[1] See Appendix for biographical sketch.

[2] There is no mention of it in the leading demography texts.

[3] On some calculators, this might be displayed as 2.56E6. Note that E is not related to the constant 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 Word. It is no longer a live mathematical worksheet.

[7] This refers to built-in procedures. In most cases, extensions or workarounds are possible. In R. for example, there is a program that facilitates the use of engineering notation, including in tables. https://cran.r-project.org/web/packages/docxtools/vignettes/numbers-in-engineering-format.html

**Appendix – Biographical Sketch:**

Thomas K. Burch is Adjunct Professor, Dept. of Sociology, University of Victoria [Canada] and a Regional Associate, Center for Studies in Demography and Ecology, University of Washington. He is Professor Emeritus, Western University, London, Ont., where he served from 1975 to 2000, working to develop a Ph.D. program in social demography. Before joining the faculty at Western, he taught at Marquette Univ., Milwaukee, Wisc. And Georgetown University, Washington, D.C. He has been a visiting professor at the Univ. of California, Berkeley and at the Univ. of Rome, and a guest researcher at the Max Planck Institute for Demographic Research, Rostock, Germany. From 1970 to 1975 he was Associate Director, Demographic Division, The Population Council, New York, NY.

His main areas of research and publication are fertility, marriage and divorce, and household and family. Late in his career he focused on a methodological assessment of demography as a science, with particular attention to theory and models. A recent summary of this work is *Model-Based Demography: Essays on Integrating Data, Technique and Theory*, Max Planck Institute for Demographic Research and Springer Open, 1918 [open access]. In a first-year university course in organic chemistry, the author used a slide-rule for all calculations. As a graduate student in the late 1950s, he used mechanical calculators that would add, subtract, multiply and divide – only – with no tapes or printouts. Calculations involving logarithms, exponentials or roots required consulting large reference books of tables. The computer revolutionized demographic calculation.