How do I know the significant numbers

Significant places

Why are significant digits important?

There are rules and conventions for the form and presentation of scientific calculations[1][2][3]. This also includes the correct spelling of figures and measurement results. Measurement results of physical quantities always contain a certain uncertainty, they are not arbitrarily precise. In scientific papers or protocols, a measurement result is therefore always given together with its uncertainty. Either through the absolute uncertainty, such as $ 1.50 (6) \ \ text m $ or $ 1.50 \ \ text m \ pm 0.06 \ \ text m $ or by specifying the relative uncertainty such as $ 1.50 \ \ text m \ cdot (1 \ pm 4 \ text%) $. The accuracy of the result, i.e. its quality, can only be judged on the basis of its relative uncertainty, because the size of the numerical value is irrelevant for this. Because the measurement of a large number with 0.1% uncertainty is much more precise than that of a small number with 10% uncertainty, although the latter has a smaller absolute uncertainty. In sample calculations, exercises and textbooks, an abbreviated notation is used and the accuracy of the number is expressed simply by the number of digits written, e.g. 1.50 m. By omitting the third and all further decimal places, it is clear that these are unknown and the last decimal place is rounded. Rounding makes a number less accurate and creates an uncertainty. The notation 1.50 m contains an uncertainty due to the rounding, as does the specification of a measurement result using $ 1.50 \ \ text m \ pm 0.06 \ \ text m $ or $ 1.50 \ \ text m \ cdot (1 \ pm 4 \ text%) $. The main difference is that the uncertainty is not explicitly given. Nevertheless, it must be taken into account in calculations! The known digits of a number are called their significant digits. Numbers may only contain significant digits. If one does not pay attention to significant digits, one feigns a smaller or a greater precision of the number than is actually available. This contradicts good scientific practice!

Understanding question 2: Sort the following numbers a = 243 ± 1 and b = 0.11 and c = 17.4 · (1 ± 1%) according to their accuracy (greatest accuracy first)!
a > c > b! The relative uncertainty of a is 0.4%, at b the omitted position could e.g. also be a 4, which results in an uncertainty of 4%, and at c she is 1%!

Rounding Numbers

In science, the rounding of a number is not done arbitrarily, but according to fixed rules, e.g. in DIN 1333[4] are described.

Rounding types

There are two types of rounding: commercial and mathematical. The commercial rounding is learned at school: The rounding point is w. If z the digit of the number at the point w is, the following number determines what with z happens: If a digit <5 follows, remains z equal (round down), followed by a digit ≥ 5, then becomes z increased by 1 (round up). The only difference to mathematical rounding is that a 5, provided it is only followed by zeros, is always rounded to the next even number.

Example: The number 15.7450 should be placed in the tens w = 0.01 (hundredths) rounded. The number there is z = 4, followed by a 5th commercial rounding results in 15.75. Mathematical rounding is 15.74.

Specifying the rounding point

For specifying the rounding point w The following relation applies: The tens place w, to which a number must be rounded, results from the uncertainty u the number through the borders u/30 < wu/ 3. A tens digit is e.g. 10 or 0.01 or 0.0001.

Example 1: Find the correct rounding point
The calculated number 0.815 has the uncertainty u = 0.06. Then u/ 30 = 0.002 and u/ 3 = 0.02. The rounding point w must lie in between and amount w = 0.01. This means that the number has to be rounded to hundredths and is 0.82.

Uncertainty through laps

The other way around, however, it also means that a rounded number inevitably implies an uncertainty u with 3w < u < 30w is afflicted. If the number is rounded to the second decimal place, it has an uncertainty u between 0.03 and 0.3. Without further information, this cannot be restricted any further. However, this uncertainty must be taken into account in calculations with this number. This is done through the correct choice of the significant digits!

Number formats and their significant digits

Numbers can be given in different formats. The most important Number formats are:

  1. Decimal notation, e.g. $ 0.074 $ with places before and after the decimal point (074)
  2. Exponential notation, e.g. $ 7.4 \ times 10 ^ {- 2} $ with mantissa (7.4) and exponent (−2) to base 10
  3. Fraction, $ \ frac {74} {1000} $ with numerator (74) and denominator (1000)

For the specification of numbers with uncertainty one uses the exponential notation as standard in science and only for "handy" numbers (approx. Between 0.001 and 1000) also the decimal notation. Whole numbers that have zeros at the end must be given in exponential notation so that the uncertainty of the number can be clearly read. In exponential notation, the symbol × must be used as a multiplication symbol, the point · is not permitted[3]. Special forms of the exponential notation are the scientific notation, in which the exponent is chosen so that the mantissa has only one non-zero digit before the decimal point, and the technical notation, in which the exponent must be divisible by 3. You can use both.

Fractions are only used for exact numbers (e.g. for angles in radians $ \ frac \ pi 2 $) and usually do not contain any uncertainty.

Counting the number of significant digits

In exponential notation, the number of significant digits is indicated exclusively by the mantissa. The procedure is the same for mantissas and "pure" decimal numbers: Read the number from front to back and look for the first digit that is not zero. These and all subsequent digits, including any zeros, are significant digits. Their number is the number of significant digits. This means that when counting significant digits, the zero has a special position: If there are one or more zeros at the beginning of the number ("leading zeros"), these are ignored. Leading zeros do not count as significant digits! On the other hand, zeros at the end are included, because they determine the rounding point and determine the precision of the number. The position of any comma is meaningless. Likewise the size of an exponent in exponential notation.

The number of significant digits of a decimal number or mantissa is the number of the specified digits minus the leading zeros!
The number of significant digits of a number in exponential notation is the number of significant digits of the mantissa!
Example 2: Reading off significant places
The decimal number 0.00740 contains the digits 7 and 4 and 0 after the three leading zeros. These are three digits and the number therefore has three significant digits. In exponential notation, it could be written as $ 74.0 \ times 10 ^ {- 4} $ or $ 7.40 \ times 10 ^ {- 3} $ or $ 0.740 \ times 10 ^ {- 2} $. In all cases it also has three significant digits. The position of the comma is irrelevant for the count! Like 7.4, 74 has two significant digits! Likewise, 0.74 and 0.074 and 0.0074 etc. has two significant digits, because the leading zeros do not count! In contrast, the numbers 7.40 and 0.740 and 0.0740 and 0.00740 each have three significant digits, because the zeros at the end after the 4 count!

Why is that? Because leading zeros only determine the size or order of magnitude of the number, just like the exponent in exponential notation. However, significant digits are a measure of accuracy, i.e. the relative error of a number! Let us consider the number 1.00 with three significant digits as an example! It has the uncertainty $ \ pm 0.05 $, i.e. its relative error is $ \ pm 5 \ text% $. In order to indicate this uncertainty or the relative error, it is unnecessary and insignificant to know which physical quantity is indicated in which unit with the number. It could be a length in kilometers: L.1 = 1.00 km. But also a length in meters: L.2 = 1.00 m. L.2 is smaller than L.1, but not known in detail. The relative error is the same in both cases. We can also L.1 express in meters L.1 = 1,00 × 103 m. This also does not change the relative error, only the order of magnitude of the number.

Understanding question 4: Sort the four numbers a = 503 and b = 0.0076 and c = 100,0 ×103 and d = 7 according to the number of their significant digits (smallest number first)!
d < b < a < c! d has a, b has two, a has three and c has four significant digits!

Difference between decimal places and significant places

The accuracy of a number must not depend on the selected number format. The number of decimal places, on the other hand, depends directly on the number format in which the numerical value is specified, i.e. whether it is represented in decimal or exponential notation and which exponent is selected. Decimal places and significant places are therefore not necessarily the same! In fact, the number of significant digits in a number determines how many decimal places you can write in the selected number format!

Example 3: Specifying a number with a specified number of significant digits
The calculation result 70.449 should be expressed with three significant digits. The decimal notation would be 70.4 and contains one decimal place. The exponential notation in "Scientific Notation" would be $ 7.04 \ times 10 ^ {1} $ contains two decimal places. The exponential notation in technical notation "would be $ 0.0704 \ times 10 ^ {3} $ contains four decimal places.

Determination of significant places of calculation results

Even calculation results should only contain digits that are significant, i.e. their value is clearly derived from the given numerical values. To do this, you have to determine the number of significant digits in the result and round the result appropriately. You have to do this yourself, because computers and calculators interpret every number input as exact. You interpret an entry of 7.4 as 7.400000 .... That is why they give way too many digits for a result and we have to determine the number of significant digits ourselves. There are the following simple rules that can ultimately be traced back to error propagation:

Products and quotients

A product or quotient has as many significant digits as the factor with the fewest number of significant digits!
Example 4: product and quotient
Calculate $ s = \ frac 1 2 a t ^ 2 $ with $ a = 1.5 \ \ text {m / s} ^ 2 $ and $ t = 1.2345 \ \ text s $! Calculation result: $ s = 0.913654 \ \ text m $. The fraction 1/2 is exact, a has two significant digits, t has four significant digits. The result must have two significant digits and is correct: $ s = 0.91 \ \ text m $.

Sums and differences

A sum or difference of two numbers is to be rounded to the smallest tens, which is still given in all summands!
Example 5: sums and differences
Calculate the sum L. the following lengths: $ L_1 = 1.16 \ \ text m $, $ L_2 = 0.0244 \ \ text m $, $ L_3 = 12.1 \ \ text m $. Calculation result: $ L = 13.2844 \ \ text m $. L.1 goes to 0.01, L.2 goes to 0.0001, L.3 goes to 0.1. Only 0.1 is given in all summands, so the result is to be rounded at 0.1 m and is correct: $ L = 13.3 \ \ text m $.

Intermediate results

Given numbers and intermediate results should always be included in every calculation without rounding and with the maximum number of digits (machine accuracy)! Only for the indication of results is it rounded to the significant digits!

The best way to calculate numerical values ​​is with the help of a spreadsheet (e.g. the free Libre Office Calc). This means that the numbers entered and the invoice always remain transparent and traceable, and interim results are automatically received with machine accuracy. This is definitely preferable to calculating with a pocket calculator! If a calculation with the pocket calculator is unavoidable, the calculation must be repeated several times in order to exclude input errors. If interim results have to be noted, these should then contain at least two more digits than there are significant digits in order to keep rounding errors small.

Example 6: intermediate results

Calculate the constant acceleration a in m / s2 and from that also the speed v in km / h for a car that moves from a standing start t = 7.5 s the distance s = Traveled 86.2 m! From $ s = \ frac 1 2 a t ^ 2 $ it follows $ a = \ sqrt {\ frac {2 s} {t ^ 2}} $. This gives $ v = a t $. The conversion to km / h is done using $ v = v_ {m / s} \ times 3.6 \ frac {\ text {km / h}} {\ text {m / s}} $. Inserting the numbers with full accuracy (machine accuracy) results in $ a = 1.750682 \ \ text {m / s} ^ 2 $ and thus $ v_ {m / s} = 13.130118 \ \ text {m / s} $ and $ v = 47.268425 \ \ text {km / h} $ (see Figure B6). The results must be rounded to two significant digits because t has only two significant digits. They are $ a = 1.8 \ \ text {m / s} ^ 2 $ and $ v = 47 \ \ text {km / h} $.

If you are wrong with the rounded value of a calculates, the result is instead $ v_ {m / s} = 1.8 \ \ text {m / s} ^ 2 \ cdot 7.5 \ \ text s = 13.5 \ \ text {m / s} $ and $ v = 48.6 \ \ text {km / h} $. The wrong lap leads to the wrong result $ v = 49 \ \ text {km / h} $.
Understanding question 5: What is the result of the following calculation: 4 (1.14 × 10-2) + 12,2 = ?
The result is 12.2! Because only the final result 12.2456 may be rounded! From the rule of sums it follows that 0.1 must be rounded to the tens. If you add the factor 4 (1.14 × 10-2) = 0.0456 rounds to a significant digit before the addition, this creates an avoidable rounding error!

General calculations

In order to determine the significant digits in other mathematical operations, one would, strictly speaking, have to know the uncertainties of all numbers and apply Gaussian error propagation, from which the aforementioned rules for multiplication and addition result. For example calculations and exercises it is sufficient to give results with the number of significant digits that the most inaccurate given number has that goes into the calculation.

Example 7: Arbitrary calculations
A force has the amount F. = 17.3 N and is at the angle θ = 24 ° to the x-axis in the xy-plane. Calculate the components of the force and give the vector $ \ vec F $! The components are $ F_x = F \ cos (\ theta) $ and $ F_y = F \ sin (\ theta) $. The vector is $ \ vec F = \ left (\ matrix {F_x \ cr F_y} \ right) $. Inserting the numbers yields F.x = 15.804336 N and F.y = 7.036544 N. The angle has only two significant digits. Therefore the result is $ \ vec F = \ left (\ matrix {16 & \ text N \ cr 7.0 & \ text N} \ right) $.

Wrong choice of significant digits

One of the most common beginner mistakes is giving a result with far too many significant digits. And when results are "necessarily" given with the correct number of significant digits, the equal sign is often replaced by ≈. This is a very good indication of the abdominal pain that beginners experience when they abbreviate a calculator display. The professional recognizes the layman immediately. If you want to teach physics, you have to master the handling of significant digits. Otherwise he gives about the same impression of his competence as a prospective German teacher who does not master the correct handling of how and as or the same and the same!