## Friday, February 1, 2013

### Day 13 - Fundamentals of Physics - 1.7 - Significant Figures

You are welcome to check out my previous blogs within the series “Fundamentals of Physics

When speaking about significant figures we are entering to and exposing the world of uncertainty within science. When making measurements, there is always going to be a level of uncertainty, as a degree of error. As appose to counting objects where everyone would count the same number, when making a measurement there is no such thing as an EXACT measurement - thus, when making measurements we do so to an accepted level of accuracy / precision. The level of precision is limited by the measuring tools and equipment as well as human error, and in some cases precession is compromised based on practical decisions made in consideration of the specific measurement, where there could have been a more accurate and precise measurement by using technology that can provide such accuracy, but it was unnecessary for the specific experiment / project. Once the level of accuracy is accepted, experiment can be held many times in order to decrease the level of error, using statistical measures.

"Significant figure" is related to the value received by the measurement, and specifically refers to the digits, within the measured value, that are certain and the first uncertain digit as well, in other words - a digit is said to be a significant figure if it is either known with certainty or if it is the first estimated digit in a measurement.

Let me explain with a simple example, when measuring the length of you desk, you may use a measuring tape as a practical tool to do so, you will notice that your measuring take is marked by line and numbers, indicating the length in which ever unit you measuring tape is aligned with. If it is a metric measuring tape you will a lines marked 1 centimeters apart with a number to indicate how many centimeters you are measuring, though it's not all, you will see that each centimeter is divided to 10 smaller parts, called millimeters, and are marked by 9 small lines -this is as far as the division goes, which means that you are limited in regards to the level of precision you are able to reach within measuring your desk, you cannot measure it in accuracy greater than the level of millimeters, which implies that the millimeter is your limit of certainty, thus, the first digit of uncertainty.

Putting what we have here all together, if you measured your desk to be 75.5cm long, due to the uncertainty you should write it as (75.5 +/- 0.1)cm, indicating that your level of uncertainty was limited at the scale of millimeter (0.1 cm), this maintains the integrity of you measurements as it incorporates it's level of certainty, you can then say that your measurement has 3 significant figures - 2 of which are certain (there is a certainty about the measuring of 75 cm) and 1 of which is uncertain as the actual exact length of the table could be anywhere between 75.4 cm and 75.6 cm.

if I'm using a measuring tape that is divided to millimeters, I am limited within my measurement as I cannot measure anything smaller than a millimeter , thus, the accuracy I can reach will be +/- 1mm, in other words, which ever value I measure it can be more or less 1 millimeter. And so I must add this range of error into the result if I want to express the precision I had throughout my experiment.

When expressing the value of your measurement the amount of significant figures should be the same no matter which unites you chose to express it by, you could write it out as 0.754m, or 75.4cm, and as you can see in both cases you have 3 significant figure. But, zeros can be tricky within this, have a look, if you would have found that the desk's length is 100cm,but due to the absence of a decimal point the amount of significant figures is vague, you can only guess if the level of certainty was in the scale of millimeters or centimeters, it is left unclear. You could also express the length of your desk as 1m, though you see that in such writing you only have one significant figure, yet you have used the same measuring tape, and thus, you should express the same level of certainty within your answer, thus, within respecting this point, it has been agreed to write it as such 1.00m where the additional zeros are there as significant figures.

This bring me to the point of scientific notation, where it has been agreed that when expressing values in any units you may chose you will write it out in such a way that the number is expressed as a number between 1 - 10 times an exponent of 10, so if you want to express 100cm with a level of certainty of 1mm, you will write it as 1.000*10^2, this way you maintain the zeros in the relevant spaces to express the level of certainty, as the significant figures and thus maintain the integrity of the measurement. The scientific notation is also good for when speaking of really large or extremely small values, where it makes it easier to clearly see the power of magnitude of the value as they are all standardized to being a number between 1 and 10, times the specific power of 10.

I will not go into the specific details of this point in regards to making calculations and maintaining the level of uncertainty as there are a few mathematical laws you must be aware of, though, the principle is that by the end of the calculation you must have a figure that expresses the level of precision within your measurement. I will place below the links I have used to gather the info for this blog, you can check it out if you want to know more.
So there we have it, significant figures as indication of level of certainty, and scientific notation as a standard way of expressing values that allow clear and easy access to plainly see the significant figures, direct calculation and are easily comparable.

What I found surprising while investigating this point in preparation for writing my blog, is that the significant figures includes not only the digits of certainty, but the first digit of uncertainty as well - I would have expected that the significant figures would represent only that which science is certain of, but I was wrong, and it's actually a cool point to consider, because it implies that the uncertainty is as significant as the certainty.

Within our process we find ourselves facing a lot of uncertainty, yet we tend to want to walk only in the known paths, were we experience ourselves as certain, where we don't have to face the "dangers" of the unknown, though it is within the levels of uncertainty where we can expand in and actually find the "missing pieces" of ourselves, the pieces we have suppressed to such an extent that they are unknown to us.

We must realize that, just as science improves it's technologies to be able to reach higher levels of precision, so can we -whenever we reach a "dead end" in our self investigation we must know that we can go deeper and reach higher levels of precision and specificity through asking ourselves questions, and allowing ourselves to self honestly open ourselves up to ourselves in self intimacy, to reveal ourselves to ourselves. Though the ability / skill to go deeper and reach the specifics of ourselves, goes hand in hand with the will and dedication to actually investigate those hidden areas within ourselves, areas we have kept in the dark from ourselves due to self judgment and shame - thus, through the process of self forgiveness we allow ourselves to release ourselves from the self judgment and shame and allow ourselves to face the points as they are, as we accepted and allowed them to exist within us and to direct us throughout our lives.

Another point this brings up within me, that I find interesting, is that, generally speaking, the word "significant" implies something large and important, where as we see that it actually refers to the small and large values, equally. This is to remind us that we exist both in the large scale as the general constructs or patterns we exist as, and the smallest of the details as they are the building blocks we have built the constructs / patterns of ourselves with - for me, I have been resisting for a long time to face myself within the specifics, as I wanted to only look at the larger scale and only investigate the general pattern, but that is self manipulation, because even though I exist as a general pattern, this character / pattern I have created myself as was built and designed through the accumulation of many small in seemingly insignificant memories / events / interpretations / experiences, and thus, when investigating myself within any pattern, I must walk through the resistance of looking at all the details and allow myself to face myself within them, because as they say "god is in the details"- we exist in the details of ourselves, and thus, in order to walk a process of self correction we must investigate the most seemingly insignificant and minute points, within realizing that they are in fact equally significant as the large scaled patterns. Further more, not only are they significant due to being the building blocks of how we created ourselves as who we are as the patterns and constructs we exist as, they are actually the key to the next layer of specificity - as we improve our level of precision by walking step by step through the levels and layers of ourselves, having to walk passed all the layers along the way.

There are no short cuts. Thus, every dead end we reach we will initially resist, though we must then find it within ourselves to walk through the resistance and investigate the specific details of this new and unknown realm of ourselves, we have entered into, to then expose the next level of precision as we come closer and closer to ourselves in oneness and equality.

http://www.pipeline.com/~merlin/signif.htm
http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_SignificantFigures.xml

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