In the previous video, we were introduced to the basics of scientific notation. In this video, we look at scientific notation in more detail and specifically look at representing very large quantities using positive exponents and very small quantities using negative exponents. First, however, let's have a quick recap.
Scientific notation is a system of expressing measured or counted quantities based on powers of 10. In the previous video, we were able to show that the number 326,400 can be represented in scientific notation as 3.264 by 10 to the 5. The 3.264 is called the coefficient of the number, and the 10 to the 5 is called the exponential part of the number. More specifically, the 10 is the base of the exponential, and the 5 is called the exponent. As this table shows, exponents that are positive represent numbers that are greater than 1. Now if we start with the number 10 to the 0 where the exponent is actually 0, we see that this number actually represents the value of 1 since 10 to the power of 0 is equal to 1. So if we have a number represented like this in scientific notation, 3.800 by 10 to the 0, this actually represents 3.800 times 1 which we know just equals the value 3.800. Now something to note here is that the decimal place in the coefficient has not moved when we represent this number in its decimal form and this is because the exponent is zero and the decimal place does not move when the exponent is zero.
Now if we look at an exponent of one i.e 10 to the power of one this number actually represents the value of 10 since 10 to the power of 1 is equal to 10. So if we have a number represented like this in scientific notation 3.800 times 10 to the 1. This actually represents 3.800 times 10 which we know just equals the number 38. This time we note that the decimal place in the coefficient has moved one place to the right corresponding to an exponent of plus 1. The positive value of the exponent tells us to shift the decimal place to the right and the magnitude of 1 tells us how many places to move it. 10 to the power of 2 or 10 squared is we know equal to 10 times 10 or 100. So if we have a number represented like this in scientific notation 3.800 times 10 to the 2, this actually represents 3.800 times 100 which we know equals 380. In this case the position of the decimal place in the coefficient has moved two spaces to the right in the decimal form of the number corresponding to the exponent of plus 2. Again the positive value of the exponent tells us to move the decimal place to the right and the magnitude of 2 tells us how many places to move it. 10 to the power of 3 or 10 cubed is we know equal to 10 times 10 times 10 or 1000. So if we have a number represented like this in scientific notation 3.800 times 10 to the 3, this actually represents 3.800 times 1000 which we know equals 3800. In this case the position of the decimal place in the coefficient has moved three spaces to the right in the decimal form of the number, corresponding to an exponent of plus three. And again, the positive value of the exponent tells us to move the decimal place to the right, and the magnitude of three tells us how many places to move it. So hopefully you're now getting the idea.
If we now look at one of the most important quantities in chemistry, Avogadro's number, which is the quantity of the mole, and has the value... of 6.022 times 10 to the 23. This is an incredibly large quantity, an incredibly large value. And if we write this number out in full decimal form, this is what it looks like.
And it's as if we move the decimal place in the coefficient 6.022. to the right by 23 places. So we're able to convert from the exponential form of a number to the decimal form of a number by seeing how many places we have to move that decimal place. And when we have a positive exponent, as in this case, it means we move the decimal place to the right. We can do the reverse as well.
We saw in the previous video that there are 167 billion trillion molecules of water in a teaspoon of water. To convert this huge number into scientific notation, we first identify 1.67 as being the coefficient. Remember, the coefficient only has one number to the left of the decimal point, and that number is never zero.
Then, in effect, what we do is count how many places we have to move the decimal place to get to the end of the number. It turns out that in this case, we have to move the decimal place by 23 places to regenerate the original number. So this number. 167 billion trillion is best expressed in scientific notation as 1.67 times 10 to the 23. So exponents that are positive represent numbers greater than one, but exponents that are negative represent numbers less than one, as this table here shows. But we're not talking about negative numbers, we're talking about numbers between zero and one.
We're talking about fractions. So again, We see when 10 is raised to the power of 0, that just corresponds to the number 1. So a number expressed like this, 1.9 by 10 to the 0, is actually equal to 1.9 times 1, which just equals 1.9. And, as before, the position of the decimal place does not move when we have an exponent of 0. 10 to the minus 1, however, represents 1 over 10 to the 1, which is 1 over 10, or 1 tenth, or 0.1 in decimal form. So if we have a number represented like this in scientific notation, 1.9 by 10 to the minus 1, this actually represents 1.9 times 0.1, which equals 0.19.
In this case, the position of the decimal place in the coefficient has moved to the left by one place in the decimal form of the number, corresponding to the exponent of negative 1. In this case, the negative value of the exponent tells us to move the decimal place to the left and the magnitude of 1 tells us how many places to move it. 10 to the minus 2 actually represents 1 over 10 to the 2 or 1 over 10 squared which is 1 over 100 which has the value of 0.01. So the number 1.9 times 10 to the negative 2 actually represents 1.9 times 0.01 which equals 0.019. And again, notice how the decimal place in the coefficient has moved two places to the left, corresponding to an exponent of negative 2. The negative number tells us we move the decimal to the left, and the value of 2 tells us how many places to move it.
And finally, 10 to the minus 3 represents 1 over 10 to the 3, or 1 over 10 cubed, which is 1 over 1000, which in decimal form is 0.001. And so the number 1.9 by 10 to the minus 3 in scientific notation represents 1.9 times 0.001, which is 0.0019. And in this case, the decimal place in the coefficient has been moved three places to the left, corresponding to an exponent of negative 3. So again, hopefully you're getting the idea now with respect to how to deal with negative exponents. You may recall from the previous video that the mass of a uranium atom is this many kilograms. To convert this number into scientific notation, we first write down the coefficient 3.95, and we identify this as the coefficient by ignoring all those initial zeros until we come to the first non-zero number, and we put the decimal place after that first non-zero number, remembering that the decimal point in a coefficient can only have one number to the left of it, and it can never be zero.
We then count backwards or we count to the left to see how many places we have to move the decimal place of the coefficient to regenerate the original number. In this case we have to move the decimal place 25 places to the left. Moving a decimal place to the left represents a negative exponent so this quantity is best expressed as 3.95 times 10 to the minus 25 kilograms.
A very very small number it's still a positive number it's still greater than zero. but it's much much less than one.