measurements and scientific notation all right so as we've discussed previously chemistry is quantitative and since we're going to be dealing with quantities we want to learn to write numerical values in convenient notation so we're going to talk today about two different kinds of notation one is standard notation and the other one is scientific notation now quantities have two parts there's the number part and then there's the unit part now the number part tells us how many that's what we're going to talk about today we're going to look at units later on so standard notation this is something that you are probably already very familiar with so these are just the standard you know ways to write numbers in everyday life they can have decimals so we can see that they can have commas to separate out the zeros like an 18,000 or just the number 25 that's a standard notation and it can be cumbersome in science because you need a large number of zeros to place nonzero numbers in the proper position and that happens often enough that we need an easier way to write really large and really small numbers conveniently and so that's where scientific notation comes in now scientific notation is basically an expression of a number using powers of 10 and so we use this to express numbers that have many zeros so so for instance so here's a power of 10 there's 10 and then it's to the zero power anything that the zero power is one 10 to the first power is 10 10 to the second power is 10 times 10 which is 100 10 to the third power is 10 times 10 times 10 which is a thousand 10 to the fourth power is 10 times 10 times 10 times 10 equal to 10,000 and then we could keep on going 10 to the fifth power is 10 times 10 times 10 times 10 times 10 or a hundred thousand and so basically that number here that is raised to a power that's called the exponent and that exponents value is always going to be equal to the number of zeros in the number expressed in standard notation okay now we can also use standard scientific notation to write really really small numbers and we're going to use negative exponents for that so 10 to the minus 1 is 0.1 okay or 1/10 10 to the minus 2 0.01 or 1/100 to the negative three point zero zero one or one one thousandth and you can see it follows the same pattern except it's a fraction now the negative exponent implies a decimal number less than one and so basically same thing here the value of the exponent is equal to the number of zeros in the denominator in the fraction okay so get familiar with those and then I'll show you a little bit easier way to deal with this later on okay so how do we write scientific notation so the first thing we want to do is write our number so that it has a value between 1 and 10 and and this is called the coefficient so we're gonna write the first non-zero digit then we're gonna write a decimal place a decimal point and then we're gonna write the rest of the digits and so this is called the coefficient then we're gonna figure out how many powers of 10 we need to multiply it by to make that number into the original number and then finally we are going to multiply that written number the coefficient by the proper powers of 10 so here's an example so we're going to write 70 9345 in scientific notation so here's our number and we're gonna write it here's the coefficient now now what I like to do is count so I'm gonna pretend there's a decimal place here and I'm going to move the decimal place to the left until I get to you know this between one and ten so our coefficient so one two so one two three four okay so we have seven point nine three four five times ten to the fourth power so see what I did there so no matter what this is I'm gonna pretend there's a decimal place there one two three four seven point nine three four five times ten to the fourth power and that is our number in scientific notation so we're going to do a few more examples later on in the presentation for small numbers we use the same process but now that power of 10 is negative okay so here is zero point zero zero zero four one one and we want to write this in scientific notation now we have a decimal place here see the decimal place between these two zeros that zero doesn't matter but I'm going to move the decimal place now one two three four until I get to a number that's between one and ten okay and since I moved it one two three four places to the right that is my exponent so four point one one times 10 to the negative four now this just refers these extra zero digits at the end or beginning of a number are generally not included so that would be this guy right here but these are important because these three zeros show the relative magnitude of that number so one two three four we have our coefficient and we've moved it to the right and so we have a negative exponent now this slide basically just puts in words what I just did on the previous slide because this is the way that I like to write small numbers into scientific notation so basically we're going to count the number of decimal places it takes to move the decimal two after the first non-zero number okay and if we move that decimal to the right that means that our exponent is negative so one two three four and we're going to get four point one one times 10 to the negative four okay so now it's practice time so we're going to write these numbers in scientific notation so take a few minutes write them out give it a shot even if you're not sure that's right just try it and then up and then we'll check the answers in just a second okay so let's start with three hundred and six thousand now that's a big number it would be nice to write that in scientific notation so what I'm going to do is I'm going to pretend that there's a decimal place here after this last zero and I'm going to move the decimal place to the left and I'm going to move it until I get to a number between one and ten okay so one two three four five so three point zero six is now our coefficient and I moved it one two three four five places so my my power of 10 is five and it's positive five because we want to take this little number and make it into this big number as it was originally so the number in scientific notation is three point zero six times ten to the fifth power so now let's do a little one okay zero point zero zero eight eight four so again we're going to move the decimal place until we get between these two eights so one two three so we have eight point eight four times 10 to the negative three we moved it three places to the right and so that gave us 10 to the negative 3 all right two million 760,000 so we're going to do the exact same thing that we did with the previous large number so I'm going to pretend there's a decimal place here after the last zero one two three four five six so we're going to have two point seven six times 10 to the six since I moved at six places one two three four five six now one thing you can do is you can always check yourself you can do 10 times 10 times 10 times 10 times 10 times 10 and then multiply that by your coefficient see if you get the the if you do then you've written it correctly so let's do one last small one so here's my decimal place and I'm going to move it till I get between these two fives so one two three four five six seven okay move just seven places to the right and we get five point five nine times ten to the negative seven all right so good job now entering scientific notation in a calculator is not necessarily trivial so you want to make sure that you have a good method for doing this you want to know how to correctly enter into your specific calculator because they're all going to be different so I'm not going to necessarily know what calculator that you have so so you want to get you know the the instructions for your calculator you can find those online and then you're gonna learn how to put that in there another thing that I like to do is I'll like let's say I have to multiply two numbers that are in scientific notation together I like to put the first number in the calculator in scientific notation and then I press equal and that stores it and then I'm gonna press x and then my second number in scientific notation and that works really really well for me but the thing that you want to do is as we go through the course and you have examples practice putting those into your calculator and make sure that you get the same the same answer because if you don't then you want to figure out what's going wrong and and so again you know let me know if you need help with that but but definitely checking the instructions or online for your calculator model that's going to help a lot and definitely practice the numbers in this presentation all right so our summary so basically we're all familiar with standard notation we probably don't call it that but that's just the normal expression of a number our scientific notation expresses a number as a coefficient times a power of 10 now that power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1 so basically less than 1 and so you want to practice converting between scientific notation and standard notation until you can do it easily