Rounding a number, what does it mean?

It is important to know some ways for a good application of numerical data when performing mathematical operations. And these paths run through the following fundamentals: significant figures, numerical rounding, precision, accuracy and uncertainty. It is about this that we set out to write this present text.

💠 Significant Figures

Imagine the length of a pen being measured with a millimeter school ruler, and, by hypothesis, that this measurement has been between the 137mm and 138mm marks, but, visually, it has been closer to 138mm, to the point that the end of the pen passed the 137mm mark by just over half a millimeter, let’s assume 0.6mm. So the measure obtained was 137.6mm, including, correct 137mm and the number 6 to represent the doubtful part that is in the space of the millimeter mark. This measure, made in this way, has four significant figures, three of which are correct and one is doubtful. It is doubtful, as it varies greatly depending on who is measuring it. It is said to be a dispersed value.

The amount of significant figures present in the data of a problem influences the presentation of calculations and results, and, if certain rules are not taken into account, it can lead to calculation errors. If data from a problem is provided, we recommend identifying its significant figures by applying the rules according to the following examples.

💠Numerical Rounding

Rounding is an arithmetic process used in the interest of approximating a number to a quantity considered more precise. It can be applied to both integer and decimal quantities.

The natural integer 57, if it were rounded, the nearest two tens are 50 and 60, as it is closer to 60 so the result is 60.

The number 1.2, if it were rounded, would be between 1.0 and 2.0, and as it is closer to 1.0 then the result is 1.0. If the number were 1.7 it would be closer to 2.0 so the result will be 2.0.

Roundings when involving business relationships or scientific interests follow a formal, or official, rule. The most usual rule is to analyze the digits that form the number or its decimal part. The main interest is to reduce the number of decimal places.

⚠️ During intermediate numerical calculations it is good practice to use the data with all its digits leaving the rounding until the end.

Calculations:

Rounding:

⚠️ Note that there are two concepts involved in this problem, rounding and significant numbers.

💠 Precision, Accuracy and Uncertainty

Physics is a science that fundamentally depends on the measurement of magnitudes, whose results can test its own theories. In this sense, precision and accuracy are fundamental elements in their daily lives.

A person who arrives at work every day about 10 minutes before office hours is an accurate person, and your colleague who arrives every day about 1 minute before office hours is not only accurate but also more accurate than the average person. first. Precision and accuracy are parameters that qualify a measurement system or process.

The measurement error is expressed by the algebraic relationship:

Measurement errors can be identified or minimized. In the case of systematic errors, as they are often intrinsic to mechanisms, these are minimized by using controlled conditions, for example, measuring the length of a metal bar always at the same temperature. In practical terms what prevails, in terms of systematic error is a more useful relationship that uses the concept of trend in which successive measurements under the same conditions tend to present values ​​within a range of values ​​and with that it is possible to establish an average value of the result.

Even if adjusted for systematic errors, random errors persist in measurement systems. The random error has its behavior explained by the Probability Theory. According to this theory, a population has a random behavior close to a Gaussian probability distribution. Within this theory the random measurement error is characterized by a standard deviation or standard uncertainty σ.

The result of a measurement is more reliable if the information is accompanied by indications of the standard uncertainty. Scientific reports, therefore, do not dispense with the treatment of measurement errors and must be considered as the object of a sampling process and therefore require the use of statistical techniques.

After the measurements and the statistical treatment, the base result and the measurement uncertainty are obtained, presented according to standardized writing.

Not all physics constants are exact values, as is the speed of light in vacuum, a large part is the result of measurements, as in the example of measuring the pen, where there is a random component that induces uncertainty, however sophisticated as the method used in the measurements is.

🎯 As a rule, the standard uncertainty value should not have, after rounding, more than two significant figures. And the base value must be rounded to contain the same number of decimal places as the standard uncertainty value