Scientific Notation Edit
Why do we use scientific notation?Edit
Scientific notation can be defined as a system to articulate small and large numbers in a coherent way. In physics, we often deal with very small quantities and very large quantities. Many people believe it is unnecessarily complex and too time consuming to constantly write out numerical values that have a large amount of digits. Scientific notation provides a succinct alternative to dealing with lengthy numbers.
Instead of writing the number .0000000000000000000007838, we can use scientific notation to write the exact same value as 7.838x10-22. This example makes it apparent that the critical purpose of scientific notation is to provide a method of representing numbers in a visual and comprehensible way, rather than in an overwhelming jumble of digits.
At first, this format may seem confusing and strange. Don’t worry! After some practice with scientific notation, it will become clear that there are countless benefits to using it. The format is undeniably more concise; it supports a deeper comprehension of the actual size of an exceptionally large/small value in relation to more familiar values.
As we now understand why scientific notation is important in the world of physics, we need to learn how to use it advantageously. In order to obtain a comfortable regents-level understanding of this system, it is crucial that we know how to convert numbers into scientific notation format. We will also need to know how to perform basic mathematical procedures (such as adding and multiplying) with numbers in scientific notation and how to express scientific notation on a graphing calculator.
Formulas and Concepts Pertaining To Scientific Notation Format:Edit
The overarching idea in regard to converting long, awkward numbers to scientific notation is to express decimal points as powers of 10.
Thus, scientific notation takes on the form:
M is often referred to as the mantissa. The mantissa must ALWAYS be greater than or equal to one and less than ten (1≤M<10). For example, the number 73000 is written in scientific notation as 7.3x104. It is not written as 730x102 or .73x105! Also, note that the exponent, represented by the variable n, must ALWAYS be an integer. Remembering these rules is crucial in converting to scientific notation correctly.
Steps for converting numbers into scientific notationEdit
- Write down the number as a decimal. (The number you want to convert to scientific notation will likely already be expressed as a decimal, but if it is a fraction or a mixed fraction, it must be changed to a decimal format.)
- Find the value of the mantissa. This value can be found by inserting a decimal after the first digit and then, dropping all the zeroes.
- Find the value of the exponent. To find this, count the number of places between where the original decimal point was located and where it should be located to make the mantissa a value between 1 and 10. (REMEMBER: if the decimal point moves to the right, the exponent is negative and if the decimal point moves to the left, the exponent is positive)
- Now that you have values for M and n, just follow the standard format of scientific notation and you are done!
Example Of Converting To Scientific NotationEdit
Convert .00000008769 into scientific notation.
- This number is already expressed as a decimal, thus it does not have to be changed in any way.
- The first digit (reminder: this does not include zero) in this number is eight, so a decimal should be inserted after the eight. Now the number is 00000008.769. Next, all the zeros must be dropped. Thus, we can conclude that the mantissa is 8.769.
- The number of places from the original decimal point to the mantissa can be determined by simply counting. We can see that there are eight places between .00000008769 and 00000008.769 by observing that there are eight numbers between the two decimal points. We must also take note that the decimal point moves to the right, so the exponent is negative. Thus the exponent value is -8.
- Finally, we have all the information to express .00000008769 in scientific notation. We know that M=8.769 and n=8. So, our final answer is…
.00000008769 can be expressed in scientific notation as 8.769x10-8
Once we are comfortable with converting numbers into scientific notation, we can learn the rules of multiplying and adding two numbers, both expressed in scientific notation.
Steps for Multiplying Two Numbers Expressed in Scientific NotationEdit
- Multiply the two mantissas together
- Add the two exponents together
- The new mantissa value and the new exponent value should be expressed in standard scientific notation format.
Example Of Multiplying In Scientific NotationEdit
Multiply 9.73x10-7 and 2.019x1015
- The product of the mantissas=9.73x2.019=19.64487
- The sum of the exponents=-7+15=8
- (9.73x10-7)( 2.019x1015)=19.64487x108. Although this value is numerically the correct answer, REMEMBER that the mantissa value must be between 1 and 10, so there is still one more step. The decimal point must be moved one space to the left and in reaction, the exponent value must be increased by one. This makes the final answer 1.964487x109.
Steps for Adding Two Numbers Expressed in Scientific NotationEdit
- Two numbers expressed in scientific notation can only be added if they have the same exponent, so the first step is to change necessary components of the problem in order to ensure that the exponent value is constant throughout the problem.
- Add the mantissas together.
- Use the sum of the mantissas as the M value and the constant exponent as the n value in order to obtain the sum in scientific notation format.
Example Of Adding In Scientific NotationEdit
- Either exponent value can be changed to equal that of the other one. In this problem, let’s convert the exponent eight in 6.2x108 to the exponent five. In order for the exponent to decrease by three units, the decimal point must be moved 3 spaces to the right. Thus, 6.2x108 can be written as 6200x105.
- The sum of the mantissas=4.3+6200=6204.3 (This is the M value.)
- The sum of 4.3x105 and 6.2x108 is 6204.3x105. In order to make the mantissa between 1 and 10, the decimal point must be moved three spaces to the left and the exponent must increase by three. The final answer in scientific notation is 6.2043x108.
Now that all of the basic rules of scientific notation and its applications have been outlined, we can learn how the scientific calculator can help us with scientific notation conversions:
Using A Graphing Calculator To Express Scientific NotationEdit
- Type in the value of the mantissa
- Press “2nd” (the button on the top left underneath “y=”)
- Press the comma button (this button says EE above the comma). After the button is pressed, a small uppercase E should appear on the screen next to the mantissa.
- Type in the value of the exponent
- Press enter (depending on the value of the number, the calculator will either automatically convert it into a regular number or keep it in its scientific notation format).
This image is from www.leydenscience.org/physics/gravitation/calc.jpg
Unit Conversions Edit
In our study of physics, we often must switch back and forth between different units of measurement. This allows us to represent different quantities in various units and types of measurements. For example, if we are working in meters, but our final answer needs to be stated in terms of centimeters, we must know how to convert back and forth between the two units. The most commonly used system of measurement is the SI system or the metric system. In this decimal system, prefixes are used to change units by powers of 10.
SI Prefixes Edit
The SI prefixes can be easily used to convert units into other units. Here is a table of the basic prefixes:
|femto||f||0.000 000 000 000 001 (10 -15)|
|pico||p||0.000 000 000 001 (10 -12)|
|nano||n||0.000 000 001 (10 -9)|
|micro||μ||0.000 001 (10 -6)|
|milli||m||0.001 (10 -3)|
|centi||c||0.01 (10 -2)|
|deci||d||0.1 (10 -1)|
|kilo||k||1 000 (10 3)|
|mega||M||1 000 000 (10 6)|
|giga||G||1 000 000 000 (10 9)|
|tera||T||1 000 000 000 000 (10 12)|
This table makes some unit conversion easy because these prefixes signify what conversion factor should be used. For example, if we want to convert 3 meters into millimeters, we know that 1 meter is 1000 millimeters, so 3 meters is 3000 millimeters.
Factor-Label Method Edit
If the conversion you are trying to do, is too difficult to simply use the prefix method, then the factor-label method is a simple way to convert units that always works. The factor-label method is a means by which one can logically change from one set of units to another. "You can factor out (multiply and divide) both the number AND the units. It has many uses. For instance, you can convert English Standard to metric units. In every factor-label problem, you must have a conversion factor. This is a fraction that is equal to one. Why is it equal to one? Because both the top and the bottom are equivalent values. Any equality qualifies (5 mi = 8 km; 100 cm = 1 m; 365 days = 1 year)." (http://www.highschoolchem.com/tut-faclab.htm)
How To Use The Factor-Label Method Edit
There are three steps in the factor label method
- State the given quantity with numbers and units
- Cancel units with the conversion factor
- Put the numbers into the conversion factor
Example Of The Factor-Label Method Edit
How many inches are in 32 kilometers?