1.5 Scientific and Engineering Notation

Transcription

1 1.5 Scientific and Engineering Notation As the names indicate, scientific notations are typically used by science and engineering notation is utilized primarily by engineering and engineering technology. The usefulness of engineering notation will become apparent in the next section in determining unit prefixes. Scientific Notation Scientific notation expresses a value as a power of 10 such that the numeric part is greater than or equal to I but less than 10. That is, the numeric part always has one and only one digit left of the decimal point. The form of scientific notation is N x 10 n where N is a number between 1 and 10, that is 1 N < 10 N must have one and only one digit left of the decimal point. The number of significant figures of N. corresponds to its original value's number of significant figures. The exponent n is an appropriate number to force the number N to be a value between 1 and 10. The exponent n can be a positive or a negative number. Example 1-14 Express the number 123 in scientific notation. You need to move the decimal point 2 places to the left to get only one digit left of the decimal point, that is, from 123 to To compensate, add 2 to the 10's exponent, that is, from 10 0 to x 10 2 Scientific notation Express 12,300,000 in scientific notation. Answer: 1.23 x 10 7 Example 1-15 Express the number in scientific notation. You need to move the decimal point 4 places to the right to get only one digit left of the decimal point, that is, from to 1.2. To compensate, subtract 4 from the 10's exponent, that is, from 100 to x 10-4 Scientific notation

2 Answer: Express in scientific notation. 1.2 x 10-7 Engineering Notation Engineering notation expresses a number as a power of 10 such that the numeric value is greater than or equal to 1 but less than 1000, and the 10 s exponent is a multiple of three. The form of engineering notation is N x 10 n, where N is a number between 1 and 1000, that is 1 N < 1000 The number of significant figures of N corresponds to its original value's number of significant figures. The exponent n must be a multiple of 3 and is an appropriate value to force the number N to be a value between 1 and The exponent n can be a positive or negative number. Example 1-16 Express the number 12,300 in engineering notation. You need to move the decimal point 3 (a multiple of 3) places to the left to get a number between 1 and 1000, that is, from 12,300 to To compensate, add 3 to the 110 s exponent, that is, from 100 to Answer: 12.3 x 10 3 Engineering notation Express 12,300,000 in engineering notation x 10 6 Example 1-17 Express the number in engineering notation. You need to move the decimal point 6 (a multiple of 3) places to the right to get a number between 1 and 1000, that is, from to 120. Note: you had to pad with a trailing 0 to create the appropriate number. To compensate, subtract 6 from the 10's exponent, that is, from 100 to x 10 4 Engineering notation

3 Express in engineering notation. Answer: 120 x Unit Prefixes Unit prefixes are a simple way to communicate very large and very small unit values. As humans we like to communicate in words. Expressing simple values like 10 seconds is very comfortable and meaningful for us; but, how about seconds? We realize this is a very small amount of time. But how small is it? It is difficult to visualize. Expressing extreme numbers in scientific notation or engineering notation is more helpful. For example, the above value would be expressed as 1 x 10-6 seconds. It is also very helpful to establish names that represent large and small unit values. These names help us more effectively visualize and communicate values. The name "micro" stands for the value Thus 1 x 10-6 second can be written as 1.0 microsecond. The term 10' is replaced directly with the word "micro". The unit prefix naming convention adopted for the SI system of units is based upon engineering notation. Look at Table 1-5. Use uppercase symbols for mega and above; use lowercase letters for kilo and below. The symbol g for the prefix micro is a Greek letter. Be careful to clearly print the prefix symbols g (long leading line) for micro (10-6 ), lowercase m for milli (10-3 ), and uppercase M for mega (10 6 ). Otherwise you will convey incredibly inaccurate information Table 1-5 SI unit prefixes Prefix Name Prefix Symbol Value tera T giga G 10 9 mega M 10 6 kilo K 10 3 unit 10 0 milli M 10-3 micro U 10-6 nano N 10-9 Pico P femto f 10-15

4 Example 1-18 Express 1 kilogram in the basic unit of grams. 1 kilogram = 1 kg Directly replace "k with its equivalent value of kg =1 (10 3 ) g = 1X10 3 g = 1000 g Note: "k" is written in the lowercase. Do not use uppercase. Express 2 megagrams in the basic unit of grams. Answer: 2 x 10 6 g Example 1-19 Express 2 milliseconds in the basic unit of seconds. 2 milliseconds = 2 ms Directly replace "m" with its equivalent value of ms = 2 (10-3 ) s = 2x 10-3 s = s Express 2 nanoseconds in the basic unit of seconds. Answer: 2 x 10-9 s Example 1-20 Example 1-21 Express 120,000,000 seconds with the proper prefix. Express in engineering notation: 120 x 10 6 S Replace 10 6 with the mega "M" prefix: 120 Ms Express seconds with proper prefix. Express in engineering notation: Replace 10-6 with the micro "u" prefix: 340 x 10 4 s 340 us 1.7 Unit Prefix Conversion Sometimes it is necessary to convert from one unit prefix to another. Remember a prefix is just a name for an engineering notation power of 10 whose exponent is a multiple of 3.

5 Example 1-22 Convert kilograms to milligrams. Move decimal point to the right in groups of 3 digits and convert to the next prefix until the number is between 1 and Start with original value kg Move decimal point 3 places right g Move decimal point 3 places right. 30 mg Note as the number gets larger, the prefix gets smaller. Answer: Convert kilograms into micrograms. 30 ug Example 1-23 Convert 4,000,000 picoseconds to milliseconds. Move decimal point to the left in groups of 3 digits and convert to the next prefix until the number is between 1 and Start with original value. 4,000,000 ps Move decimal point 3 places left: 4,000 ns Move decimal point 3 places left: 4 us Move decimal point 3 places left: ms Note as number gets smaller, the prefix gets larger. Convert 50,000 milligrams into kilograms. Answer: 0.05 kg 1.8 Unit Consistency and Conversions You must take units very seriously as noted earlier in this chapter. It is sometimes necessary to convert to other units within the same system of units or to convert a value to a different system of units. You must carry units throughout your calculations to ensure accurate units in the resulting answer. Calculations without units are not meaningful. To convert to a new unit, you must multiply by an equivalent value of 1 or a unity multiplier, for example, 60 seconds per minute. The following converts 5 minutes (min) to seconds (s): time = 60s 5 min x = 300s min

6 Notice the unity multiplier of 60 seconds per minute does not change the value of the answer. It only changes the value's units from "minutes" to "seconds". In the unity term, "minutes" is strategically placed in the denominator to cancel the original "Minutes" in the numerator. And the equivalent number of "seconds" per minute is placed in the numerator. The "minutes" cancel, and the answer is left in units of "seconds".

7 Example 1-24 You are offered a major contract at 2 cents per millisecond for 2 hours. The actual job cost is $100,000. Will you make money? Prove your answer. Use unit conversion, multiplying by a conversion factor that is an identity. Minutes are abbreviated "min" and hour is abbreviated "hr". Note: units are set up in the conversion factor so as to cancel out the unit you want to eliminate. For example, if you want to cancel out a unit in the numerator then you must multiply by a conversion factor with that unit in the denominator. 2cents 1$ 100ms 60s 60 min x x2hr = $144,000 1ms 100cents s min hr Looks like an excellent deal, a profit of $44,000 in 2 hours. Convert 3 cents per us to dollars per minute. Answer: 1.8 X 106 dollars per minute Example 1-25 Answer: 0.03 s Convert 230,000 microseconds to seconds. Use a unity multiplier to convert from us to s s 6 10 s 230,000usx = 0. 23s us Convert 30,000,000 nanoseconds to seconds.

8 Example 1-26 Convert grams to micrograms. Use a unity multiplier to convert ftom g to gg ug gx = 50ug g Convert seconds to nanoseconds. Answer: 700 ns Example 1-27 Convert 670,000,000 milligrams to kilograms. Use a unity multiplier to convert from mg to g, then another unity multiplier to convert from g to kg g kg 670,000,000mgx = 670kg 3 mg 10 g Answer: 70 us Example 1-28 Convert 70,000 nanoseconds to microseconds using two unity multipliers: (from ns to s) and (from s to gs). The next example demonstrates converting between systems of units, from metric to English. The Arecibo National Astronomy and Ionosphere Center in Puerto Rico has the most sensitive astronomical telescope in the world. The diameter of its spherical shaped reflector is 305 meters, the largest antenna dish in the world. Convert the diameter expressed in SI units of meters to the English units of feet. 100cm 1inch 1 foot 305 mx = 1000 feet 1m 2.54cm 12inch Note that each fraction is a unity and its denominator is set up to eliminate (convert) a unit.