I’m ashamed to be a senior majoring in math and I still haven’t grasped the concept of amusing significant digits! I seriously need it explained to me like I’m a child.
In: Mathematics
Okay. Let’s say you drive from New York to San Francisco… that’s about 3000 miles, right? You get there, park your car, and it rolls back an inch. So now you went 3000 miles minus 1 inch, or 2999 miles 5279 feet and 11 inches.
That’s absurd, right? Because NY to SF isn’t exactly 3000 miles… and even if it was, it’s certainly not exact to the inch. So how can you solve this?
That’s what significant digits are for. When you give somebody a number, you give them not only the value for the number, but you also tell them how well you know the number. 3000 miles isn’t wrong, it’s just not super accurate. If instead I said 2902 miles, you can infer that I know this down to the number of miles. If I said 2902.3 miles, you can infer that I know this down to one tenth of a mile.
There are rules for adding and multiplying significant digits as well. For example, if one person borrowed your car and drove “714 miles” and somebody else drove your car and drove “I dunno… 500 miles,” you can add them up and get 1214 miles… except that is implying that you know this to the mile. Really you only know this to the nearest hundred miles, so it would be more accurate to say “1200 miles,” because you added 714 and “about 500.”
Significant digits are a way of writing numbers to indicate their precision. Every digit except zero is always a significant digit. Zero is only significant if writing it wasn’t necessary to show how big or small the number is. So the zeroes in 100, 20, 0.3, 0.04 are not significant; those numbers all have one significant digit each. On the other hand the zeros in 1.0, and 20.03 are significant.
The assumed precision of numbers is plus or minus half a unit of the least significant (right-most) significant digit, e.g.
* 1 means 1±0.5
* 2.0 means 2.0±0.05
* 30 means 30±5
* 31 means 31±0.5
* 400 means 400±50
* 5600 means 5600±50
* 0.007 means 0.007±0.0005
Sometimes a number has more precision than you can show this way. For example, Everest was once measured to be 29 000 feet high, with high accuracy, but just writing 29 000 implies ±500 feet. The surveyor added two feet to write 29 002, to better indicate the precision implied.
The scientific way out of this issue is to use scientific notation which always has one digit followed by a decimal point, and then by as many extra significant digits as required. The size of the number is indicated by multiplying by a power of ten, e.g.:
* 2.9000×10^4 (the measured height of Everest)
* 1.00×10^3 (equivalent to 1000±5)
* 3.00×10^(−2) (equivalent to 0.03±0.000 05)
Usually you should retain the number of significant digits in calculations. You sometimes see journalists convert measurements in a silly way, so that “1000 miles” becomes “1609 km,” even though the second value implies much more precision than does the first. This one is a tricky case, since staying strictly with the one significant figure of the first measurement would require writing “2000 km,” which will often be misleading. In my view, adding one more significant digit in cases with just a single significant digit that’s a 1 or 2, and writing “1600 km,” is acceptable, provided that no more digits are added in subsequent calculations.
Edit: added example.
Simple explanation: write in terms of scientific notation, sig figs are the number of digits you use in the number (ignore the 10 to the power of some integer). Multiply it out and include the 0’s accordingly.
Example: 2 sig figs. 1.0×10^(-3) = 0.0010
How long does it talk you to walk to class?
If you said 10 minutes, I’d assume you were just rounding to a convenient number, and the actual time might be anywhere from 8 to 12 minutes. But if you said 11 minutes, that shows you know the duration to a higher degree of accuracy, otherwise you would have said 10. I know that it takes you 11 minutes, not 10, not 12.
That’s essentially what significant figures are. All measurements have a degree of inaccuracy, and significant figures are kind of like a guarantee, if you say 174 rather than 200 or 174.2841230, I know you mean 174 +/- 0.5 and can make assumptions based on that degree of accuracy.
One mistake students often make is overstating accuracy by using to many significant figures. Just because your calculator gives you 10 decimal places, that doesn’t mean your answer has that degree of accuracy. You need to analyze and round to the appropriate number of significant figures. Also, when you multiply two numbers together, your answer has as many significant figures as the least accurate number. If you traveled 56.7754 km in “about” 2 hours, your speed is 30 km/h. It doesn’t matter how accurately you measured the distance, if you only know the time to one significant figure, you only know the answer to one significant figure.
Significant figures are those that matter. When you drive a car the actual speed is not being displayed on the speedometer. Something close to it is. If you are driving at 35.45 mph. Your car will only display two digits (or three is you are going faster than 99 mph.)
The displayed speed has two significant figures.
The limit on significant figures reported is the least accurate measurement used. Imagine that your car gets 22.35 mpg. But you have a havd gas can that hold 5 gallons.
Saying that you are able to travel 5*22.35 =111.75 miles is kind of silly. The 5 gallon tank might have had more or less gasoline that 5 gallons. So we would only report one significant figure and we would say we can go 100 miles. The zeros to the right of the 1 are not significant, they are place holders to give you magnitude.
If you had been asked what is the average millage per gallon of your car, you might have reported two significant figures 22.35 you look at the 5 so you rough up the 0.35 to 0.4 you then round again 0.4 rounds down to zero. So 22.35 mpg is reported as 22 mpg.
Going back to the 5 gallon tank. The 5 gallons is really more like 5.49-4.50 gallons because of significant figures. Those values would be rounded to 5. Writing 5.0 makes it look more precise but it is not correct since the gas can is not that accurate.
Good for you admitting that you need help.
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