0 to #scicomm “superhero”? Seems pretty *significant* Go *figure*! You may see a lot of numbers when it comes to measuring and describing matter, but how many of those matter-describing numbers really matter? (Looking at you 3.33333…)SIGNIFICANT FIGURES are digits (whole numbers) that actually tell us info that we KNOW and you can think about it as what you can learn from a partly-filled theater row!
Imagine that a row of theater seats gets filled from the right and then you bring in a detective to try to see what you can learn from the audience members’ final positions. The 1st person sits down “blindly” anywhere so seats to her left don’t tell us anything (theater could pull back a curtain & reveal an infinite number of empty seats to her left, but it wouldn’t matter).
BUT, now when choosing where to sit, people have to make a conscious decision about how close they want to be to person to their left. So the number of empty seats between them DOES tell us something (smelliness?)
Seats to right of last person to sit down may or may not hold info. If those seats really are available, but no one sat there, that has meaning. BUT if the seats are “reserved” or cordoned off, there aren’t really any more seats available so we’re “lying” if we say there are. And if someone sits there later, it’s your job as usher to kick them out
What does this have to do with numbers and measurements? Person-filled chairs are like nonzeros & empty chairs are like 0s. Being “significant” is like being a (non-cordoned-off) chair that someone consciously did or didn’t sit in. If the chair was available and no one sat there that tells us something – those 0s *are* significant* but if that chair was cordoned off the fact that it’s empty doesn’t tell us anything – those 0s are NOT significant.
When you make a measurement, the measuring device decides where the row ends. Even when measuring the exact same thing, you’ll get different numbers of sig figs depending on the tool. Say you’re measuring something that’s *exactly* 1m long. Some instruments have really long rows so you can get lots of sig figs (like 1.00000cm). But other ones have short rows so you can only get a few (1.000m) What’s it to you?
Once you start doing calculations with those measurements, those rows can appear to get longer but it’s really just people sitting in cordoned-off seats. (Think of dividing 1 by 3….) It’s your job as a good scientist to act as an usher and kick those uninvited folks out.
So we need to know how many figures are significant. How do you know how many sig figs you have? Any NONZERO # IS significant & you start counting at 1st one you meet. 0s may or may not be significant…
If there are nonzeros on each side, the 0 must reside ✅ If the 0’s in the lead, erasing can proceed ❌ But if the 0’s at the end 👉 well, then that depends…❓If there’s a decimal preceding, that end 0 does have meaning ✅ If there’s a decimal at the rear, that 0 you hold dear✅ If there’s no decimal but you had to round, confusion might abound 😬 In such a situation, turn to scientific notation 👍
- 0’s before 1st nonzero (LEADING 0s) are NOT significant ❌
- 0’s in between nonzeros ARE significant ✅
- 0s after last nonzero (TRAILING 0s) are NOT significant ❌UNLESS there’s a decimal point ✅
e.g, 1, 01, & 001 all have 1 sig fig, but 1.0 has 2 & 1.00 has 3 👉 If I add 0s at the front, they’re just “fluff” ❌ BUT if they’re at end AND there’s a decimal point, they’re actually telling you something ✅ (on the other hand, 1, 10, & 100 all have only 1 sig fig bc there’s NO decimal point)
⚠️ EXCEPTION ALERT 👉 EXACT NUMBERS have unlimited sig figs ∞
- #s that are “defined” (like 1 yard = 3 feet)
- “counted” #s (like “6 people”)
📝 A note on SCIENTIFIC NOTATION 👉 10^something part only adds NONSIGNIFICANT 0s ❌ 👉 only count sig figs from “unique part” (e.g 9.0×10⁴, 9.0×10¹⁵, & 8.4×10¹⁵ all have 2 sig figs BUT 9×10⁴ only has 1)
WORKING WITH SIG FIGS 👉 if you divide 1 by 3 on your calculator, don’t go scrambling to write down 1.3333333333333….
If you divide 1 by 2 you get 0.5. If you divide 1 by 4 you get 0.25 & if you divide 1 by 8 you get 0.125. An answer of 0.125 might sound more knowledgeable than an answer of 0.5, but do you really know more? 🤷♀️
The more digits you include, the more it *seems* like you know 😎 BUT the amount you REALLY know is limited by amount you know about the thing you know the least about 🙄 & this is limited by your measuring tool 📏
Every measuring tool has limits 👉 If I weigh this on a “kitchen scale” it tells me it weighs 1.00g (3 sig figs). Factoring in rounding, it could “actually” weigh anywhere from 0.95-1.04g, but this scale can’t tell me bc it’s limited to hundredths place 👉 If I instead weigh it on this analytical balance, it gives me value of 0.9999 (4 sig figs) (which could actually be 0.9995-1.0004g) 👍
This # of sig figs relates to yesterday’s discussion of the difference between accuracy & precision in the context of pin the tail on the donkey.
In that case, we were dealing w/multiple “measurements” that we were averaging. If you take multiple measurements of the same thing, PRECISION tells you how close those measurements are to *each other* & ACCURACY tells you how close the average of those measurements is to the REAL value 🔑
But often when we’re measuring something in a lab, we only get “1 shot,” to measure 😬 BUT we can still provide an indication of how precise the value is 👉 in this case a measure of how confident we are in our measuring tool 👍
👉 this is where SIGNIFICANT FIGURES (sig figs) come in 👉 each # you can *confidently* measure (even 0s) IS SIGNIFICANT (tells you “real info”)), then you get 1 significant “estimate” as your last sig fig 👇
So when you pin the tail on the meniscus, how many sig figs do you report? If you’re measuring liquid w/a graduated cylinder (like a tall skinny measuring cup) & bottom of meniscus (“smile” 🙂 u measure from) is between 2 lines (aka gradations – hence “graduated” cylinder 💡), you’re sure it’s above the one 😎 so you write down that # confidently ✍️ but then you have to “eyeball” how close it is to line above 👀
You only get to write down 1 sig fig here (so if it’s between 100 & 200 you can write 150 but not 155 👉 your 👀 aren’t that good w/o lines to help out 👉 & that 0 isn’t significant, it’s just a placeholder
How close together lines are will determine how many decimal places & sig figs we can get 👉 closer the lines, the more precise you can get (assuming you’re reading the readout correctly!). Digital displays take out the “eyeballing” 😅 BUT they too have to round to their limit 👇
This analytical balance gives me more sig figs than this lab scale, so it’s more precise 🎯 BUT is it more accurate? 🤥 Not necessarily 👉what if I forgot to tare it (0 out weight of the weigh boat) 👉 I could get a really precise value ✅ but it would give me an INACCURATE weight of the solid ❌ (though it might give me an accurate weight of solid + weigh boat)
The balance has an extra aid to help w/accuracy 👉 “cage” prevents “distractions” from altering readout 👍
It’s easier to tell if something is precise than if it’s accurate bc to know if something’s accurate you need to know the real value & that’s what you’re trying to find in the 1st place 😬 👉 a good sign is if multiple different measuring tools give you the same answer 👍
Since I’m getting the “same” value from both scales, I can feel pretty sure the measurement’s accurate (assuming I remembered to tare!) 😅
So original # of sig figs (which you now know how to count 👍) comes from # you get when you MEASURE something ⏩ if you then perform CALCULATIONS w/those measured #, you need to make sure you’re not adding “fake” sig figs 👉 this is like ppl sitting in reserved seats 👉 as a careful scientist, it’s your job as usher to kick them out 👋
Speaking of ROUNDING…👇
- round at the end 👉 where you round will affect your final answer
- add 0s as “placeholders” for nonsignificant numbers you had to remove
e.g. 17 200 has 3 sig figs 👉 rounded to 2 sig figs: 17 000 👉 or 1 sig fig: 20 000 👍
⚠️ Things can get confusing if you have to round to a 0 😬 (e.g. if you round 19 900 to 2 sig figs, you have to round to 20 000 which looks like it only has 1 sig fig 👉 to avoid ambiguity, you can instead write it in scientific notation 👉 2.0 X 10⁴ 👍)
ADDITION ➕ & SUBTRACTION ➖ 👉 answer should have same # of DECIMAL PLACES as starting value w/least DECIMAL PLACES
- e.g. 10.37-1.2 = 9.2 (NOT 9.17)
MULTIPLICATION ✖️ & DIVISION ➗👉 answer should have same # of SIG FIGS as starting value w/least SIG FIGS
- e.g. 20 x 13.4 = 300 (NOT 268)
When you add or subtract, decimal *points* 👉 to your fact 👍 But multiply or divide, then you need to consider both sides 👍
Try it out for yourself in pics! 📸