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Next topic: section 2 Real Functions, page 1 Basics
We are used to present numbers in decimal form with the use of 10 digits (including the zero). This is not the result of free choice, but the sole reason is that people began to calculate using the fingers of both hands. In the future, when we will meet intelligent creatures from other places of the universe, they'll probably use a different system for presentation of numbers, suitable to their physical attributes.
Computers store data and calculate numbers in binary form with two digits (on and off), where 10 means two. Some civilizations on the earth found it easier to use binary form for fractions of unity. For example this is still in use for parts of an inch e.g. for the diameter of screws and pipes. Half of an inch shoud be expressed as 0.1 inches in binary notation and ${\scriptstyle \frac{1}{8}}$ - as 0.001 inches.
The hexadecimal form of presentation with $16={2}^{4}$ digits uses the digits a, b, c, d, e, and f after the 9. It is commonly used in software as a more convenient alternative to the binary system.
After finding out that there is nothing sacred about 10, we will continue from here on to concentrate only on the decimal presentation of numbers.
Any number presented with finite number of digits (independent of where the decimal point is) is a rational number. This is easily seen from the definition of rational numbers (for someone that already forgotten: see page 1). But each rational number cannot be expressed as a finite number of digits. For example in the case of:
$\frac{83}{35}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{2}\text{.3714285714285}\left(\text{714285}\right)\mathrm{....}$ |
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it is noticeable that although there are not finite number of digits needed for the decimal presentation, from a specific digit onwards there is an infinite repetition of a finite sequence of digits, shown in parentheses. It can be shown that this is a property of rational numbers presented in decimal form. A presentation as a finite number of digits can also be seen as a infinite repetition of zero. One can state therefore that any decimal presentation, which consists of an infinite repetition of a finite sequence of digits (including zero), from a digit on corresponds to a rational number. The opposite is also correct: that each rational number can be presented in this way.
What about irrational numbers? As a consequence of what is stated above: irrational numbers cannot be fully represented in decimal form. For example the well known number pi looks like:
$\pi =\text{3}\text{.141592653589793}\mathrm{...}$ | (1.1.4.1) |
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but cannot be presented in full. There were times when the calculation of π was a mathematical challenge and mathematicians were writing papers with more and more digits calculated, but in our age of computers, this is useless.
In empiric sciences, all the values we measure and even the constants of nature (e.g. the speed of light) have errors due to our measuring tools. The errors impose on us some restrictions on writing the expression of a value in decimal form. In a way this is very fortunate: How could we write a number with infinite number of digits?
When a numerical value is written, e.g. 345.6328951 the first digit (3) is much more significant than the following digit (4) etc.. Let's assume that we know the measured relative error of this value, say it is 1/1000 (one pro mil). This means that the error is about 0.35 and only the forth digit is affected. We don't need more than 4 (or maximum 5) significant figures (it means digits), so the value should be written as: 345.6 (or 345.63). The use of more digits is not only superfluous, but it is also wrong. The number of digits written reflects the accuracy of the value.
Zero is a legitimate digit and is included in the count of significant figures. By continuing the example, the value 345.6 is represented by four significant figures. On the other hand 345.600 is interpreted as the same numerical value, but with six significant figures, meaning much more accurate.
To summarize: We start by counting the left-most non-zero digit as the most significant figure, and we keep counting any digit to the right, even if it is zero. The result of this numeration is called the number of the significant figures and reflects the accuracy of a numerical value.
A mathematical operation on numbers can only increase the relative error, and therefore we may have to express the result with a smaller number of significant figures. For example if we have a multiplication of a number with n significant figures by another with m - , then the result should contain the smallest between n and m significant figures. As an example the circumference of a circle 2πr should be presented with about the same number of significant figures as the radius r.
An introduction to error evaluation is part of the education in the lab and we are not going to elaborate here any more. However it is very important that the final result of a numerical evaluation reflects its accuracy, by using the correct number of significant figures.
Now we come to a question that you may be already asking: What is so important in the presentation of an irrational number, if we have to limit it to a number of significant figures and it becomes a rational number. Well, you are right, it is not important! In empirical sciences we don't care and we don't know if the speed of light, measured in some particular units, is a rational or irrational number. The irrational numbers we learned about, are actually needed here only for introducing the accurate mathematical way of analysis, and for general education. We will use the notion of irrational numbers occasionally, just in order to make a point, but it does not play any role in empirical-science education and research.
It is a common practice to use the sign $\cong $ or $\approx $ (approximately equal) instead of = (equal) when a number is expressed in less significant figures. As an example the number π (1.1.4.1) can appear in the following legitimate forms:
$\begin{array}{l}\pi \approx 3.1416\\ \pi \approx 3.142\\ \pi \approx 3.14\end{array}\}$ | (1.1.4.2) |
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The numbers we use could be, in absolute value, very small or very large and not very convenient for presentation. In such a case we use a factor of 10 raised to an integer power. As an example a small number could be written as
$0.000000032865=32865\times {10}^{-12}=32.865\times {10}^{-9}$ | (1.1.4.3) |
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There are no rules about the choice of the power of 10, but it is very common to use integers having a factor of 3 (postive or negative). Of course the number of the most significant figures should be preserved.
For the case of large numbers the presentation with 10 raised to a positive power is essential. Otherwise one can end up with trailing zeros beyond the most significant figures. An example of a large number is
$32865\times {10}^{12}=32.865\times {10}^{15}$ | (1.1.4.4) |
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but not: 32,865,000,000,000,000 .
The power itself is a number. A mistake of one unit, results in an error of a factor 10, meaning an error bigger than the most significant figure. A factor of ten is commonly called an order of magnitude. It is of utmost importance that any result of a numerical evaluation should have the correct order of magnitude.
An equality between two numbers means that the numbers are the same. Sometimes one can compare quantities with different units. Then the quantities should be the same, rather than the numerical values. For instance an angle can be expressed in degrees or radians. The following equation is correct:
${180}^{\text{o}}=\pi \text{\hspace{0.28em}}\text{rad}$ | (1.1.4.5) |
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In science most of the numerical values are expressed in units and therefore their quantities should be the same in an equality. The following example compares two temperatures of different scales:
${0}^{\text{o}}\text{F}=-{32}^{\text{o}}\text{C}$ | (1.1.4.6) |
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In science it would be senseless to present any quantity without specifying its units if they exist. Keep this in mind in case of numerical calculations.
In science most of the measured quantities have not only units, but also physical dimensions. As an example length is one of the dimensions. It may be expressed in different units as meter, inch, mile etc. and we already know about comparison of different units. Another example of a dimension is time. Time also can be expressed in different units as second, hour, week, year etc. and we know how to compare time intervals given in different units.
Time and length are different entities and we cannot compare a time inteval with a length interval. An equation should have the same dimensions on both sides. It follows that we cannot add or subtract quantities with different dimensions.
If we denote t as a quantity of time, we will use square brackets ot t: [t] for the dimension of time, which will be denoted by Θ . Therefore we can write:
Θ = [t] | (1.1.4.7) |
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Let's do the same for a length l:
Λ = [l] | (1.1.4.8) |
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It is possible to multiply and/or divide quantities with different dimensions. For example the velocity v is a quantity obtained from the ratio of length to time. Therefore the dimension of the velocity is:
$\left[v\right]=\left[\frac{l}{t}\right]=\frac{\Lambda}{\Theta}$ | (1.1.4.9) |
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The acceleration a is obtained as a ratio of velocity to time and its dimension is:
$\left[a\right]=\left[\frac{v}{t}\right]=\frac{\frac{\Lambda}{\Theta}}{\Theta}=\frac{\Lambda}{{\Theta}^{2}}$ | (1.1.4.10) |
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Physics teaches us that the linear motion with constant acceleration is:
$l={l}_{0}+{v}_{0}t+{\scriptstyle \frac{1}{2}}a{t}^{2}$ | (1.1.4.11) |
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The factor ${\scriptstyle \frac{1}{2}}$ appearing in (1.1.4.11) is a pure number and is dimensionless. By use of (1.1.4.7-10) one obtains that each term in (1.1.4.11) has the same dimension Λ, as it was stated previously (one cannot add quantities with different dimensions).
After we saw an example of manipulation with the dimensions Θ and Λ we are ready to learn about the dimension of mass m:
$${\rm M}=\left[m\right]$$ | (1.1.4.12) |
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From the laws of Newtonian Mechanics follows that all the dimensions of the physics variables can be expressed by these 3 dimensions: Θ, Λ and Μ. As a consequence of this statement one can obtain in some cases physical relations just by balancing the dimensions. That will be demonstrated by an example.
The physical dimensions are very helpful when solving symbolic (not-numerical) problems in science. The final answer should have the correct dimensions and if not - one can inspect the balance of the dimensions in the equations, going backwards, till the error is found.
Example
Question. A spring, when stretched to a length l, exerts a resisting force F, according the following relation:
$$F=-kl$$ | (1.1.4.13) |
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where k is the constant of the spring. It is known that the dimensions of a force are:
$\left[F\right]={\rm M}\Lambda {\Theta}^{-2}$ | (1.1.4.14) |
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When a mass m is attached to the free end of the spring, stretched to a length l, the mass performs a periodic motion with a period of time T.
Could we find how T depends on the parameters: m, l and k ?
Answer. From (1.1.4.13) and (1.1.4.14) it follows that the dimensions of k are:
$\left[k\right]={\rm M}{\Theta}^{-2}$ | (1.1.4.15) |
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The dependence of T from the parameters in the most general case is:
$T=c{m}^{\alpha}{l}^{\beta}{k}^{\gamma}$ | (1.1.4.16) |
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where c is an unknown dimensionless constant, and the powers α, β and γ have to be found. By substituting the dimensions in (1.1.4.16) one obtains:
$\Theta ={{\rm M}}^{\alpha}{\Lambda}^{\beta}{\left({\rm M}{\Theta}^{-2}\right)}^{\gamma}$ | (1.1.4.17) |
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The balance of the dimensions in (1.1.4.17) yelds:
$\begin{array}{l}\alpha +\gamma =0\\ \beta =0\\ -2\gamma =1\end{array}\}$ | (1.1.4.18) |
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Finally from the solution of (1.1.4.18) and substitution in (1.1.4.16) one obtains:
$T=c\sqrt{\frac{m}{k}}$ | (1.1.4.19) |
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This solution shows exactly how the period T depends on m and k, and that T is independent of l. All this was obtained without solving the equation of motion.
In this section you saw important tips for self-testing the solutions of different problems. Here they are repeated for reference and another one is added at the end.
Exercise 1. Show that any number expressed in decimal form with finite number of digits, is a rational number!
Exercise 2. An evaluation of a distance outdoors is done by measuring the time t between sending and receiving the echo of a sound signal returned from a very massive rock. The time obtained is:
t = 1 min. + 4.332 sec., with an error = ±0.001 sec. |
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v = 344 m/sec with an error of ±5 m/sec |
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Exercise 3. A simple pendulum is swinging. In principle the time period of a swing T could depend on the following parameters: the swinging mass m, the length of the pendulum l, and the constant acceleration on the surface of the earth g.
Exercise 4. The force of a non-linear spring is:
$F=-q{x}^{3}$ |
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Next topic: section 2 Real Functions, page 1 Basics