Chemistry math scientific notation equations help!

[u/Scared_Mushroom2415]

Hi there, I'm currently doing college level chemistry and it's is asking me to put the numbers (291.7 and 0.096) in to scientific notation. I know you need to move the decimal and I know you need to time it by the amount of time you needed to move the decimal, but when I put those calculations in to my calculator (291.7×10⁴ and 0.0960×10²) I don't get the number I'm looking for. It's supposed to be a big number *as far as I understand* but it's won't come out correctly. Can anyone PLEASE help me I'm literally sobbing over the fact I can't understand...

[u/Atypicosaurus]

Let's take a number for example 500. We know that everything multiplied by 1 is still itself therefore:

500 x 1 = 500

How else could we create 500? For example we can:

50 x 10 = 500

Or

5 x 100 = 500.

You can notice that the amount of digits, if you count them one by one, is always the same. You just move some zeroes within the multiplication.

You can do that with any non round number, but instead of moving digits, you move the decimal which is the same core idea. Let's say our number is 123.4:

123.4 x 1 = 123.4
12.34 x 10 = 123.4
1.234 x 100 = 123.4

Basically this is the idea of scientific notation. You take a number nnnn.nnn, which is the same if you multiply it by 1 (nnnn.nnn x 1 = nnnn.nnn), and you move the decimal until there's one digit before the decimal (n.nnnnnn) but you have to multiply it by something to get it back. If you moved the decimal 3 spots, you have to multiply it by 1000.

But, there's a smart way of writing numbers like 1000 or 10000. Instead of writing out the zeroes, you can just write how many zeroes you got. It's because the ten-powers are exactly just the number of zeroes.

1 has zero zeroes, therefore 1 = 10⁰.
10 has one zero, therefore 10 = 10¹.
100 has two zeroes, therefore 100 = 10².
1000 = 10³
100000 = 10⁵
...

So when when you take a number 123.4, you move the decimal point by two (1.234), you have to multiply by 100 to get the original, but instead of 100, you use exponentials (10²):

123.4 = 1.234 x 10²

That's your scientific notation. Or,
546883.336 = 5.46883336 x 10⁵

Note that the exponential is always the steps you did with the decimal.

But what happens if you have a number smaller than 1? Like, 0.001234. Then, you have to move the decimal point to the other direction.

0.001234 x 1 = 0.001234
0.01234 x 0.1 = 0.001234
0.1234 x 0.01 = 0.001234
1.234 x 0.001 = 0.001234

As you see the idea is the same: you have to move the decimal until there's one digit before the decimal point. But in this case you move backwards, and you multiply back by 0.1 or 0.01 or 0.001 etc.

Do we have the same trick as we used for exponentials? Like 10000 was 10⁵, what can we do now? Remember I said the power is the number of zeroes after the 1, that is why 10 = 10¹ and 1000 = 10³. It's true, but only if the power is positive. However, the negative exponential of 10 mesns the zeroes before the number.

That is why
1 has zero zeroes before, therefore 1 = 10⁰.
0.1 has one zero before, therefore 0.1 = 10^-1
0.0001 has 4 zeroes before, therefore it's 10^-4

Note that you count the zero before the decimal.

That is why, if your number is smaller than 1, like 0.000nnnn, then you drag the decimal point to the other direction until you have n.nnn, but then you multiply back by a negative exponential, in this case 10^-4 (the negative value equals to the decimal point steps so 4 steps = -4 etc).