Because every science course I’ve taken so far has left me even more confused than ever.
In: 15
Really large or small numbers can be very long when written or typed, and it’s a lot easier to read a figure times a power of 10 rather than counting decimal places in many cases.
There also issues of conveying the precision of a measurement. If someone writes out “9,300,000 miles”, it’s not immediately clear if they mean “9.3 million miles”, or “exactly 9.300000 million miles”. Writing the former of the two implies that the measurement may be rounded to the nearest hundred thousand miles.
Scientific notation is used to represent really small or large numbers, or number in cases where how many digits you can be sure are accurate really matters.
To make scientific notation, you take the number – let’s say 1867 – and write it with the first digit in the ones column, and the rest after the decimal point. In this case, 1.867. Then, you multiply it by the power of ten that would give you that number – this is the number of places you had to move the decimal. In this case, you moved the decimal three places to the left – x10^3.
To give some examples of where this is useful, writing 0.0000000000782g requires counting zeroes, and leaves a lot of room for error. With a number that small, doing math with it is pretty unintuitive anyway, so changing it into the harder to visualize 7.82×10^-11 makes it a lot easier to work with without making it particularly harder to imagine. *Note – most places in the world don’t use commas in their large or small numbers, so all the zeros blur together like that
It can also be helpful in places where significant figures, or how many digits you know from data and how many just showed up in the math, matters. If you measure that you have 10.0 grams of something, and you know that 1/3 of that by weight is green, then the answer that you’d get is 3.33333333333333333…g of green. However, your scale’s only so accurate. Maybe you have 10.00478083984g total grams. You have no way of knowing that anything past 3.33 is actually correct, so you don’t write this.
In a case where you end up with a very large number, like 1 888 967, and you only know that the first three digits are accurate, how do you write that? If you write 1 890 000, it still kind of implies that you’re sure of those zeroes. The best way to do it is 1.89×10^6
Its for really, really big and very, very small numbers which you will get to when you are dealing with atoms and chemistry as well as astro-physics.
Like the number of atoms in 1 mol of an element. Its a 6 with 23 zeros after it. Who has time to write that? Its 6.02*10^23. Much easier to write that than to write 620,000,000,000,000,000,000,000.
The basic principle of scientific notation is just keeping things relatable and simple by shifting decimal points.
For instance 1000 has 3 zeros to the right of the 1, thus it can be written as 1*10^(3).
0.001 on the other hand has 3 Zeros to the left of the 1 and thus can be written as 1*10^(-3). This is because division and multiplication by 10 simply moves the decimal point when using the decimal system.
Why is that useful? Why not just write 1000 or 0,001? why not 1k for 1000?
Well such a convention makes working with very small or very large numbers way easier.
For instance if you’re working with gravity you’ll have to use the mass of planets, stars or even heavier objects. The Earth has a mass of 5.97*10^(24)kg, this means it is a 5.97 and now you shift the decimal point 24 places to the right. That’s way too large to write every time you want to calculate something.
Same goes for very small objects, if you wanted to work with the mass of a proton you’d have to deal with a value of 1.6*10^(-27)kg. Do you really want to write that many zeros in front of the value?
On top of that it makes calculations way simpler because one can leave everything comparable by using the same units. See how one value is very large, one is very small but both are still given in kg? This makes harder calculations simple using rules of exponents.
For example imagine you had two values:
A: 2 quintillion
B: 8 septillion
Now calculate A*B.
Do you know what 2 quintillion times 8 septillion is? You probably get the 16 right immediately but what about the units? What the heck is a quintillion times a septillion?
Instead of breaking our heads on the naming, just convert them to scientific notation.
A: 2*10^(18)
B: 8*10^(24)
Now calculating A*B is as easy as doing 2*8 and 18+24, the result is: 16*10^(42)
You could now shift the decimal by one to the left and write 1.6*10^(43) instead and be done (although 16*10^(42) would also be perfectly valid, so this step is optional).
What’s easier to write? 5430000000 or 5.43×10^9 ?
Once numbers get really big or really small, we don’t want to write all of those zeros. This is also why we have prefixes for units. 5.1nm is 5.1×10^-9 m it’s just an extension of scientific notation.
It also makes calculation by hand much simpler. 6.9×10^8 * 1.2×10^-3 / 3.4×10^4 is the same as (6.9*1.2/3.4)×10^(8-3-4) or 10^1. So now I have an expression with 2 digit numbers and a single exponent.
It’s also very easy to see how many significant figures a number has. 1.2×10^7 has 2 significant figures, but 1.200×10^7 has 4 significant figured. If we had the number 12000000, according to sig fig rules, it has 2, but there’s no way to express that it’s 4. Significant figures are important for notation how precise a calculated number is.
This reminds me of the question at the dinosaur museum asked by a visitor to one of the staff.
“How old are these dinosaur bones?”
And the answer the staff member gave was
“66,000,004 years”
“How do you know the exact answer, to the year?”
“Well, when I started working here four years ago, the bones were 66 million, so now they must be 66,000,004 years”
This is why scientific notation is important, because we know the actual answer is only significant to two digits, 6.6×10^7 so the actual age of the bones could be anywhere from 65-68 million years, and isn’t known to be exactly 66,000,004 years old.
Just like how there are 8 billion people on the planet. When your sister’s child is born, that doesn’t make the population of the planet 8,000,000,001 when the actual population is 7.753×10^9
Why it’s necessary. Here is the mass of the sun:
198850000000000000000000000000 kg
Number of atoms in a gallon of water:
300000000000000000000000000
The Width of an atom
0.000000001 m
And I might be wrong. I might have miscounted the number of zeros. There’s a lot there.
It’s far easier to say the mass of the sun is 1.9885×10^30 That means to take 1.9885, and move the decimal right 30 times before you read it out. No accidentally dropping a zero.
I could write that same number many ways, depending on what I want to compare it to.
19.885 x10^29
198.85 x10^28
etc.
The earth has a mass thats ~6×10^24 kg.
So if I write the mass of the sun as
19885×10^24 cmpared to earths
………6×10^24 it’s easy to see how much bigger the sun is compared to earth. Sort of like saying 3 “million” rather than 3,000,000
One reason scientific notation is useful because counting digits is hard.
Lets consider 50 million. That’s “5” followed by 7 zeros. “50000000”.
That looks an awful lot like 5 million, “5” followed by 6 zeros. “5000000”.
Honestly, can you count those zeros easily?
But if I write 5.0×10^7 then you don’t have to count the zeros because you know there are seven of them.
And if I write 5.0×10^6 then you know there are 6 zeros.
This is helpful even if they aren’t all zeros. Consider a number like 15 million. That’s 1.5×10^7 Instead of 7 zeros, it’s 7 digits after the first digit.
Good answers all around but one addition: Scientists don’t like scientific notation either. Generally, they’ve found units that make sense when working with whatever scale they need.
Thus astronomers describing stars don’t give miles or meters for distance but light years. Far away galaxies tend to get MLY – million light years. Same idea for particle physicists, but on a much tinier scale.
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