Using Significant Figures and Scientific Notation

Using Significant Figures and Scientific Notation When making a measurement, a scientist can only reach a certain level of precision, limited either by the tools being used or the physical nature of the situation. The most obvious example is measuring distance. Consider what happens when measuring the distance an object moved using a tape measure (in metric units). The tape measure is likely broken down into the smallest units of millimeters. Therefore, theres no way that you can measure with a precision greater than a millimeter. If the object moves 57.215493 millimeters, therefore, we can only tell for sure that it moved 57 millimeters (or 5.7 centimeters or 0.057 meters, depending on the preference in that situation). In general, this level of rounding is fine. Getting the precise movement of a normal-sized object down to a millimeter would be a pretty impressive achievement, actually. Imagine trying to measure the motion of a car to the millimeter, and youll see that,  in general, this isnt necessary. In the cases where such precision is necessary, youll be using tools that are much more sophisticated than a tape measure. The number of meaningful numbers in a measurement is called the number of significant figures of the number. In the earlier example, the 57-millimeter answer would provide us with 2 significant figures in our measurement. Zeroes and Significant Figures Consider the number 5,200. Unless told otherwise, it is generally the common practice to assume that only the two non-zero digits are significant. In other words, it is assumed that this number was rounded  to the nearest hundred. However, if the number is written as 5,200.0, then it would have five significant figures. The decimal point and following zero is only added if the measurement is precise to that level. Similarly, the number 2.30 would have three significant figures, because the zero at the end is an indication that the scientist doing the measurement did so at that level of precision. Some textbooks have also introduced the convention that a decimal point at the end of a whole number indicates significant figures as well. So 800. would have three significant figures while 800 has only one significant figure. Again, this is somewhat variable depending on the textbook. Following are some examples of different numbers of significant figures, to help solidify the concept: One significant figure49000.00002Two significant figures3.70.005968,0005.0Three significant figures9.640.0036099,9008.00900. (in some textbooks) Mathematics With Significant Figures Scientific figures provide some different rules for mathematics than what you are introduced to in your mathematics class. The key in using significant figures is to be sure that you are maintaining the same level of precision throughout the calculation. In mathematics, you keep all of the numbers from your result, while in scientific work you frequently round based on the significant figures involved. When adding or subtracting scientific data, it is only last digit (the digit the furthest to the right) which matters. For example, lets assume that were adding three different distances: 5.324 6.8459834 3.1 The first term in the addition problem has four significant figures, the second has eight, and the third has only two. The precision, in this case, is determined by the shortest decimal point. So you will perform your calculation, but instead of 15.2699834 the result will be 15.3, because you will round to the tenths place (the first place after the decimal point), because while two of your measurements are more precise the third cant tell you anything more than the tenths place, so the result of this addition problem can only be that precise as well. Note that your final answer, in this case, has three significant figures, while none of your starting numbers did. This can be very confusing to beginners, and its important to pay attention to that property of addition and subtraction. When multiplying or dividing scientific data, on the other hand, the number of significant figures do matter. Multiplying significant figures will always result in a solution that has the same significant figures as the smallest significant figures you started with. So, on to the example: 5.638 x 3.1 The first factor has four significant figures and the second factor has two significant figures. Your solution will, therefore, end up with two significant figures. In this case, it will be 17 instead of 17.4778. You perform the calculation then round your solution to the correct number of significant figures. The extra precision in the multiplication wont hurt, you just dont want to give a false level of precision in your final solution. Using Scientific Notation Physics deals with realms of space from the size of less than a proton to the size of the universe. As such, you end up dealing with some very large and very small numbers. Generally, only the first few of these numbers are significant. No one is going to (or able to) measure the width of the universe to the nearest millimeter. Note This portion of the article deals with manipulating exponential numbers (i.e. 105, 10-8, etc.) and it is assumed that the reader has a grasp of these mathematical concepts. Though the topic can be tricky for many students, it is beyond the scope of this article to address. In order to manipulate these numbers easily, scientists use  scientific notation. The significant figures are listed, then multiplied by ten to the necessary power. The speed of light is written as: [blackquote shadeno]2.997925 x 108  m/s There are 7 significant figures and this is much better than writing 299,792,500 m/s. Note The speed of light is frequently written as 3.00 x 108  m/s, in which case there are only three significant figures. Again, this is a matter of what level of precision is necessary. This notation is very handy for multiplication. You follow the rules described earlier for multiplying the significant numbers, keeping the smallest number of significant figures, and then you multiply the magnitudes, which follows the additive rule of exponents. The following example should help you visualize it: 2.3 x 103  x 3.19 x 104   7.3 x 107 The product has only two significant figures and the order of magnitude is 107  because 103  x 104   107 Adding scientific notation can be very easy or very tricky, depending on the situation. If the terms are of the same order of magnitude (i.e. 4.3005 x 105  and 13.5 x 105), then you follow the addition rules discussed earlier, keeping the highest place value as your rounding location and keeping the magnitude the same, as in the following example: 4.3005 x 105   13.5 x 105   17.8 x 105 If the order of magnitude is different, however, you have to work a bit to get the magnitudes the same, as in the following example, where one term is on the magnitude of 105  and the other term is on the magnitude of 106: 4.8 x 105   9.2 x 106   4.8 x 105   92 x 105   97 x 105or4.8 x 105   9.2 x 106   0.48 x 106   9.2 x 106   9.7 x 106 Both of these solutions are the same, resulting in 9,700,000 as the answer. Similarly, very small numbers are frequently written in scientific notation as well, though with a negative exponent on the magnitude instead of the positive exponent. The mass of an electron is: 9.10939 x 10-31  kg This would be a zero, followed by a decimal point, followed by 30  zeroes, then the series of 6 significant figures. No one wants to write that out, so scientific notation is our friend. All the rules outlined above are the same, regardless of whether the exponent is positive or negative. The Limits of Significant Figures Significant figures are a basic means that scientists use to provide a measure of precision to the numbers they are using. The rounding process involved still introduces a measure of error into the numbers, however, and in very high-level computations there are other statistical methods that get used. For virtually all of the physics that will be done in the high school and college-level classrooms, however, correct use of significant figures will be sufficient to maintain the required level of precision. Final Comments Significant figures can be a significant stumbling block when first introduced to  students because it alters some of the basic mathematical rules that they have been taught for years. With significant figures, 4 x 12 50, for example. Similarly, the introduction of scientific notation to students who may not be fully comfortable with exponents or exponential rules can also create problems. Keep in mind that these are tools which everyone who studies science had to learn at some point, and the rules are actually very basic. The trouble is almost entirely remembering which rule is applied at which time. When do I add exponents and when do I subtract them? When do I move the decimal point to the left and when to the right? If you keep practicing these tasks, youll get better at them until they become second nature. Finally, maintaining proper units can be tricky. Remember that you cant directly add centimeters and meters, for example, but must first convert them into the same scale. This is a common mistake for beginners but, like the rest, it is something that can very easily be overcome by slowing down, being careful, and thinking about what youre doing.

Top-level Accommodations for Students in the USA

Top-level Accommodations for Students in the USA Best Students USA Accommodations Apparently, you are convinced that college accommodation is definitely one of the worst things you should survive while obtaining your degree. Besides doubtful neighbors, you will face living in a new space, which most of us imagine like small, uncomfortable room, where one is stuck for three or five years. Thus, most students try to hire flats or live at their parent’s place, which can be not very convenient due to additional time and transport expenses. Students’ accommodation is usually situated on the campus territory, so living there you have distinctively more time for sports, library attendance, writing essays or entertainment with your friends. So, what is it better to do in such situation and how to cope with this difficult choice of place for living? Before preferring any of your options concerning living conditions, we recommend you not to haste and find out everything about the accommodation at your college. You will be exceedingly surprised, but we want to astonish you with the fact that accommodations can be even architectural masterpieces. So meet these well-planned buildings, which dismantle all stereotypes. Baker House This building is designed and built for Cambridge students in Massachusetts. The project of the Baker House is executed by the Finnish architect Alvar Aalto. The architect embodied his ideas to provide a view out over the Charles River for each student and located the Baker house along the river. Besides, Alvar Aalto tried to avoid making typical and ordinary room design. Thus, he created 22 bedrooms on each floor. These bedrooms are saturated with individuality due to different shapes. The building has facade, made from red brick and fascinates all passersby by its look. That is so great for students to live in such an outstanding accommodation, which can become not only the place for living, but one of the vivid and bright recollections, concerning their years at the university. Peabody Terrace This students’ housing is located also in Harvard, Massachusetts and was designed by an architect Josep Luis Sert. Being the dean of the architecture school at Harvard University, Josep selected several room designs and arranged them into sections, creating terraces. The building has complex and peculiar design, but there can be hardly any students, who will not appreciate living at such astonishing place. Architects emphasize it outstanding design and ideological commitment and it will be an exciting experience for students to spend their time at this building. Noyes House This student dormitory is located in New York at the territory of the Vasar College campus. This building was designed by Eero Saarinen in slick crescent-shaped form. It was built in 1958 and symbolized the new breath to the residential life of the students. Though its architecture has respectful age, at those times it was a real breakthrough in the design of such buildings. Noyes House gives house for 178 students and is not a simple accommodation. It represents the first, progressive steps in the area of architecture masterpieces in regard to college and campus buildings. Olympia Avenue This is totally innovative residence hall for students, located in the Washington State. This building combines great interior design and superb interior planning, which presupposes lounges, kitchens, laundry facilities and studies, located on each floor. Except modern design this hall offers a lot of friendly features – storm water collection, geothermal heating and cooling, site restorations by means of natural landscaping and vegetation. So along with the living function Olympia Avenue executes the educational function, teaching its dwellers to take care about the environment. These students’ accommodations prove that there are always exceptions from the rules and college dormitories can be a great place for living. Just imagine how much friends you will make and what a great time you will spend, living in one of such dormitories at the campus area. Treat it as one of the adventures of your life and enjoy the best years at your college.