1.1.1 State and compare quantities to the nearest order of magnitude.
An order of magnitude indicates the relative size of a quantity. Long and complicated numbers can be simplified or estimated through the use of this method. For example, the number 200 is in the order of magnitude of 10^2 (100). When estimating orders of magnitude -- similar to rounding -- anything below 5 is the lower order of magnitude while anything above or equal to 5 is the higher order of magnitude. In this case, 500 and anything above would be in the order of magnitude of 10^3 (1000). Some common orders of magnitude include:
Practice questions
1. Estimate the number of seconds in a human "lifetime".
Answer: 70yrs = 2.2 * 10^9 sec, 100yrs = 3.1 * 10^9 sec, 50yrs = 1.6 * 10^9 sec, therefore the estimate of the order of magnitude is 10^9 sec.
1. Estimate the number of seconds in a human "lifetime".
Answer: 70yrs = 2.2 * 10^9 sec, 100yrs = 3.1 * 10^9 sec, 50yrs = 1.6 * 10^9 sec, therefore the estimate of the order of magnitude is 10^9 sec.
1.1.2 State the ranges of magnitude of distances, masses and times that occur in the universe, from smallest to greatest.
1.1.3 State ratios of quantities as differences of orders of magnitude.
Orders of magnitude can be compared in order to create a ratio. For example, the size of an atom is 10^-10m whereas the size of a proton is 10^-15m. The difference between these two values is 10^-5, so the atom is 10^5 (100,000) times bigger than a proton. The ratio of the atom to the proton would be 1:10^-5.
Practice questions
1. Estimate the ratio of
a) the mass of an electron to the mass of a human
b) the radius of earth to the size of the universe
Answers:
1. a) compare 10^-30 and 10^2, 10^32 is the difference hence 1:10^32 b) compare 10^7 and 10^26, 10^19 is the difference hence 1:10^19
1. Estimate the ratio of
a) the mass of an electron to the mass of a human
b) the radius of earth to the size of the universe
Answers:
1. a) compare 10^-30 and 10^2, 10^32 is the difference hence 1:10^32 b) compare 10^7 and 10^26, 10^19 is the difference hence 1:10^19
1.1.4 Estimate approximate values of everyday quantities to one or two significant figures and/or to the nearest order of magnitude.
Significant figures are the individual digits in a numerical value. For example, the number 20 has two significant figures. One important rule regarding significant figures is that when there are zeroes preceding the number (ie. 0.020), they are not counted, hence the value 0.020 would still have two significant figures. Zeroes after the number (ie. 20.00) or between values (ie. 201) are always counted.
In order to express significant figures as orders of magnitude, one should take all of the numbers following the amount of significant figures given and convert them to a power of 10. For example, the number 0.0030 would be 3.0 × 10^-4; the zero after 3 remains because it was significant in the original value. Sometimes a significant figure amount will be given to you, for example, "56,000 (3 s.f.)" would convert to 560*10^2 because 560 has three significant figures and that was the requested amount in the original question.
Sometimes numbers need to be rounded in order to fit a significant figure amount. If the amount of significant figures required is two and your answer is "56,000", you should rewrite it as "5.6 × 10^4". Similarly, if your answer is "0.0560823" and you require two significant figures, you should round it down to "0.056" or rewrite it as "5.6 × 10^-2".
In order to express significant figures as orders of magnitude, one should take all of the numbers following the amount of significant figures given and convert them to a power of 10. For example, the number 0.0030 would be 3.0 × 10^-4; the zero after 3 remains because it was significant in the original value. Sometimes a significant figure amount will be given to you, for example, "56,000 (3 s.f.)" would convert to 560*10^2 because 560 has three significant figures and that was the requested amount in the original question.
Sometimes numbers need to be rounded in order to fit a significant figure amount. If the amount of significant figures required is two and your answer is "56,000", you should rewrite it as "5.6 × 10^4". Similarly, if your answer is "0.0560823" and you require two significant figures, you should round it down to "0.056" or rewrite it as "5.6 × 10^-2".