Transcript for:
Ultrasound Physics Math Concepts

hi learners it's m with sano nerds today's video is going to be on unit 1 formulas and mathy stuff there are several formulas that you will be shown throughout your studies of ultrasound physics understanding the formulas will be key to understanding the physical concepts of ultrasound so we're going to work on manipulating formulas and how to break them down to better understand relationships the formulas that you will have are going to have a variable on one side so the answer and then everything else is going to be equal to the answer on the other side here's a really simple example about what i mean we have our answer a is equal to the variables b divided by c now sometimes you may be given the numerical value for a and for c and you have to solve for b so to do that we need to apply some simple algebraic principles to the formula to transpose it so we can rearrange the formula to solve for b or solve for c depending on the information that we have what's really important though about manipulating formulas is that you must always keep the relationships between all the variables the same no matter how it's presented and to do that basically need to perform the same mathematical operation to both sides of the equation let me show you what i mean by that so i told you that we have a equals b over c and what happens when we want to solve for b well we need to figure out how to get b by itself so we're going to need to get rid of c and we can see right now that c is dividing b so we basically need to do the opposite of that so we'll multiply by c to get rid of the division by c but if we do it to one side we have to do it to both sides now we have c times a equals b times c over c and here's one of the first algebraic principles that you need to know whenever you have the same variable on either side of a division line those are going to equal 1. and we know that 2 divided by 2 is 1 30 divided by 30 is 1 so that checks out c divided by c should equal 1. so now we have c times a equals b times one and another principle that we know is that b times one or a variable multiplied by 1 or any number multiplied by 1 does not change so b times 1 equals b and in the end we end up with c times a equals b so we have moved from a equals b divided by c to c times a equals b those are all the same relationships this actually puts us in a really good spot though now to solve for c so we left off with c multiplied by a equals b and now we want to get c alone so we have to do something with the a since a is being multiplied by c we want to do the opposite of that and in math the opposite of multiplying is dividing so we're going to divide by a but remember we got to do that to both sides so the a's are going to turn into a 1 and we can basically get rid of that and we end up with c equals b divided by a so in the end we had a equals b divided by c c times a equals b and c equals b divided by a now if you're ever curious did i do that math right we can double check it say that a equals 10 b equals 20 and c equals 2. well now we can plug these numbers in and see if our math checks out so let's look at our first one we have 10 would be equal to 20 divided by 2. that one checks out 20 divided by 2 is 10. let's look at the next one c times a equals b so that would be 2 times 10 equals 20. that one also checks out i'm sure you can guess what's going to happen with the last one but just to make sure we have 2 equals 20 divided by 10 and indeed it does so all three of our equations check out the math works for all of them we have not changed the relationships to anything because we did the same mathematical operations to both sides of the equal sign to isolate our variables go ahead and pause the video because we are at our first practice moment go ahead and see if you can work through the problem on your own and when you're ready unpause and i'll show you the answer so here are the answers how'd you do well the question is asking us to rearrange the formula for wavelength so it shows us that wavelength equals c over f c is propagation speed f is frequency and lowercase lambda represents wavelength so we want to solve then for both frequency and for propagation speed this actually really follows our pattern that we just saw a equals b over c so why don't we head over to the board to see how we do this so we have lambda equals c divided by f and we're just going to do the same thing we need to multiply both sides by f to isolate c and that is how we are going to get c by itself propagation speed then equals the frequency multiplied by the wavelength or lambda again this sets us up really nicely to be able to solve for f then we just need to divide both sides by lambda and we'll get that frequency equals c divided by wavelength now that you know how to rearrange formulas to solve for the variable that you want we also need to discuss how to describe the relationships within those formulas that's actually going to be a really big theme of ultrasound physics we're going to ask you when x increases what happens to y if b decreases what happens to a so we first need to talk about how things can be related and then we'll talk a little bit more about how to recognize those relationships there are five relationship statuses number one is unrelated two items that have no association with one another we then also have related these are going to be two items that are connected but really no specified relation third we have directly related and directly proportional this means that there are two items that are related so that when one increases the other also increases they go in the same direction now that direction doesn't always have to be increased they can also both decrease fourth we have inversely related and inversely proportional so this is two items that are related so that when one increases the other decreases or if one decreases together increases and lastly we do have a special inverse relationship called a reciprocal reciprocal are when two factors are multiplied together they're going to equal the number one now there are two relationships that are especially important to the ultrasound physics world and those are going to be inverse relationships and direct relationships for example we're going to learn that when frequency increases period will decrease this means that frequency and period are inversely related they're going in opposite directions another example is when power increases intensity increases note that the arrows are both going in the same direction that means these are directly related oftentimes we just include arrows for the increased decrease as a quick shorthand one thing that might be kind of confusing though for students is talking about the relationships in that increasing and decreasing terms when we say increasing or decreasing we are referring to the numerical value that represents that parameter but what the variable is doing in the real physical world might actually be a little bit different or referred to differently so for example i told you that when frequency increases period is going to decrease what's actually happening is that when frequency gets higher the periods get shorter and i've got an example below that we've got frequency equals 5 megahertz increases to 10 megahertz so we can see that the numerical value has changed 5 to 10 that's an increase what's actually happening though is that the frequency has gotten higher that is how we would explain it same with period then we have the period at 0.02 microseconds decreasing to 0.01 microseconds although the period is decreasing another way to say it is that the period has shortened the time has gotten less let's go ahead and take a look at the direct example then when we say power increases intensity increases but what's really happening is that when power gets stronger intensity gets stronger and again with the example we have power moving from 5 watts to 10 watts we can see that the numerical value has changed we can also see that the intensity 2.5 watts per centimeter squared increased to 5 watts per centimeter squared the numbers have increased but what's actually happening is that both of those bigness parameters are getting stronger so when you're thinking about relationships within the formulas i encourage you to understand what's actually happening in the physical world it'll help you to kind of conceptualize or visualize what's actually happening with these parameters go ahead and pause we're going to take another moment to practice relationships and formula you're going to decide if statements are directly related unrelated or inversely related welcome back here are the answers to our first practice the amount i read my textbook and my physics grades are directly related in theory if you read your textbook more your physics grade should go up however if you don't read your textbook then your physics grade would probably go down so the idea is that they are both increasing getting better doing more or they're both getting worse decreasing or getting poorer the second one asks us to compare the number of trees in the forest and my house address well first off we're not even mentioning what forests we're talking about and could you really count all the trees hard to say for sure but that has absolutely nothing to do with your house address so these are unrelated next up we have the time spent on instagram and my cell phone data usage so if you're not around wi-fi and you're just doom-scrolling on instagram you're going to end up using a lot of your data so the more time you spend on instagram the more cell phone data that you're going to use so your time has increased your data usage has increased they are going in the same direction therefore they are directly related next we have the number of purchases made and the store's inventory if we have a lot of people coming into a store and purchasing items and taking them out of the store that means that the store is giving up inventory or their inventory is going down so we've increased purchases which will decrease the inventory these are inversely related lastly we have the number of days with cold temperatures and the number of days that i turn the furnace on if we have a lot of cold weather we're going to turn the furnace on more so increased days of cold temperature means increased days of furnace opposite of that then if there are less days or decreased days of cold temperatures then we are going to decrease the days that we turn the furnace on so again because the ideas are going in the same direction these are directly related so now that we know the relationships are how do we recognize these relationships and formulas well it really all has to do with how the formula is set up first i want to go over some terms though to help us kind of explain the rules as we go through them the first thing we need to know is that the variables are called factors so we have a times b and b divided by c these are all factors within the formula when you multiply two factors together the answer is called the product and when you divide two variables the answer is called a quotient before moving on i just want to highlight that when we are talking about relationships and formulas we're gonna focus on only changing one factor at a time and discussing how that change in the factor affects the product or the quotient we also need to be able to do it the opposite way if we change the product or the quotient how does that affect each of the factors so there are two rules when we talk about relationships in formulas rule number one the factors are directly related to the product so i want to bring up our simple formula a multiplied by b equals c so those are a and b are factors c is our product and then i also have the numbers plugged in just so we can kind of see what happens when we change our variables first we're going to talk about a if we increase a then we should see an increase in c so a went from 10 to 20 c went from 20 to 40. the other thing i want to note here is that these are directly proportional note that 10 to 20 is a doubling and 20 to 40 is a doubling now the same is going to be true for b we increase b 2 to 6 and then we saw increase from 20 to 60. now in this case note that 2 to 6 is a increase by a factor of 3 or a triple and 20 to 60 increase by a factor of 3 or a triple now those examples were looking at increases we're going to see a very similar thing happen with decreases note here that we have a decreasing from 10 to 5 therefore c is going to decrease 20 to 10. same idea 10 was halved so 20 is halved same thing will happen with b we decrease from 2 to 1 therefore c will decrease 20 to 10. now rule number two has a couple parts to it let's look at the first part the quotient is directly related to anything above the division bar which is known as the numerator so let's bring back our simple equations we have a equals b divided by c and again plugging in some simple numbers just to check our math so let's talk about b first if b increases and remember we're not changing c around right now we're just talking about changing one variable at a time so b increases 20 to 40. then a the quotient should also increase and we see that it does 10 to 20. note again these are moving in the same proportion b was doubled so a is doubled same idea when we look at the decreasing part of it if we decrease b 20 to 10 we should see a decrease in a and we do 10 to 5. so the numerator or b in this example is directly related to a let's look at the second half of that rule though rule number two the second part says that the quotient is going to be inversely related to anything below the division bar which is known as the denominator let's go ahead and bring back our formula c in this scenario is the denominator so a and c are going to be inversely related let's take a look so now we're saying that if c increases then a should decrease because they are going to go in opposite directions so indeed c increased from 2 to 5 and we see a decrease in a 10 to 4. the same is true then if we decrease c so we go from 2 to 1 we should see an increase in a 10 to 20. so again to reiterate the quotient the answer to a division problem is directly related to anything above the division bar or the numerator and inversely related to anything below the division bar or the denominator all right we're at our second practice for relationships and formulas go ahead and pause the video and when you are ready come on back and we'll take a look at the answers so for the second practice you are asked to look at some formulas and discuss the relationships within them so this one is giving us the formula for the nyquist limit so nyquist limit equals half multiplied by prf so nyquist limit is the product half and prf are the factors and we want to know in the first one what happens to the nyquist limit when the prf increases and then also what happens when it decreases well we know that when factors increase that products will do the same thing so if prf increases then the nyquist limit should also increase if the prf decreases then the nicos limits decrease the next box asks us what happens to the prf when the nyquist limit increases or decreases now we really didn't talk about this as much in the examples but it could be inferred that if the factors and the products are directly related to one another that if the product changes that it's going to affect the factors in the same way and that's exactly what happens so if the nyquist limit were to increase then we would see an increase in prf if the nyquist limit were to decrease then we would expect a decrease in the prf as well because they are directly related now the next formula that we're given is intensity is equal to power over area squared so this is asking now what happens to intensity if the power increases and decreases well power and intensity in this type of formula setup are directly related the numerator is directly related to the quotient so if power increases then intensity needs to increase if power decreases then intensity decreases next up we're asking what happens to intensity if area increases or decreases well now area the denominator is inversely related to the quotient so when area increases intensity should decrease when area decreases intensity is going to increase and lastly kind of like we did with the last one in the nyquist limit curious what you came up with for what happens to intensity how does that affect power and area well if the intensity increases that means that either power increased or area decreased and if intensity decreases that means that power must have decreased or area increased now before we put formulas aside my last bit of information for you is a study tip what i really suggest you to do is to create a formula sheet this will mean to take a document take a piece of paper put it in the front of a notebook in front of your folder and this is just going to be a piece of paper that you're going to add to every time we come across a new formula in our studies when you come across a new formula write the formula down make sure to define each of the variables sometimes they're not super obvious remember c represents propagation speed you might not remember that so make sure that you're defining your variables and including units for all of your formulas then i would write down all of your relationships within the formula and transpose your formulas to solve for all the variables so in the event that you're given values for two of the variables you can just plug your numbers in and solve for the variable that you don't have i strongly believe that the more you work with the formulas and the more that you are able to manipulate them and use them in real numbers and plug things in and practice it the more you're going to understand the relationships that they represent next up we've got some mathy things to talk about now i hear it all the time i'm not very good at math is physics going to be hard well we really can't get away from math when we are studying physics but the cool part is is that this class is not only physics but it's also instrumentation so yes we are going to spend some time on physical properties learning formulas and manipulating those formulas but we're also going to learn about our machines and transducers and how the image is actually created so do not despair not all of this class will be math and formula based but yes there is a lot of it now another thing that i want you to know is that we can't use calculators on our boards and so the math on your tests and your national board exams is usually very minimal you do need to know how to add subtract multiply and divide basic numbers you should also know how to raise numbers to powers and convert units all without a calculator now we spent the first portion of this lecture talking about the relationships that are found within the formulas and that's really what i want you to focus on when we are talking about the formulas because those are the types of questions that you're going to get on your boards they're going to say variable a increase by a factor of 2 what happens to the quotient or variable c decreased by a factor of 4 what happened to the quotient so we need to know what increasing and decreasing by a factor means but then we also need to know those relationships so we can report back what actually happened let's take a look at an example using our simple formula so let's say that our question tells us that variable b increases by a factor of two to increase by a factor of two simply means to multiply by two and if we know that it's multiplying by two we then need to look at the other relationships that are affected by b increasing and we know that b and a are directly related so if b increases by 2 or is multiplied by 2 or increase by a factor of two then a also has to increase by a factor of two or be multiplied by two so a question that might go along with this would be something like variable b increases by a factor of two what happens to a and your answers might be doubled halves decreases by a factor of four quadrupled and so you're going to show the examiners that you understand the relationship between b and a that they are directly related proportionally related and you're going to say that a doubled what happens then if the question then tells you that b decreases by a factor of three when you decrease by a factor that means to divide by three or divide by the factor so b decreases by a factor of three which means it's a third stronger than it was and again b and a are directly related so a also needs to decrease by a factor of three or b divided by three or become one third as strong they need to do the same thing let's take it a step further then let's take a look at what happens when we talk about increasing and decreasing c by a factor so again let's say that our problem tells us that c increased by a factor of two so c increases by a factor of two that again means simply to multiply it by two well if c increases by a factor of two or doubles what happens to a well a and c are inversely related so that means that a must decrease by a factor of two be divided by two or become half as strong they need to do opposite things so this is where knowing if we are increasing means to multiply decreasing means to divide by that factor and then understanding the relationships between our factors and the answers to easily answer the questions on the test all right so let's move on to some of our basic mathy things that we need to get out of the way first up we've got units so i mentioned in the study tip that i want you to make sure that you're when you write your formulas down you should also include the units and that is because units define numerical values i've got the number 360 here but you have no idea what i'm talking about until i add minutes now we know that we're talking about a unit of time or if i add miles well now we're talking about a distance if i add square yards that means we're talking about an area where cubic centimeters tells us a volume you have to define your numbers and if your answer does not have a unit or a percentage it's wrong until you define it now some common dimensional units that we are going to use in ultrasound include length now length can be anything that's a distance unit such as centimeters meters feet mile meters and so on we use length typically to measure anatomy or pathology and we'll measure the length of an object in three planes usually in the transverse plane longitudinal plane and anterior posterior plane however we can also measure the distance around a circle and that's known as a circumference that is the length around a circle another type of unit that we might use then is area and area are going to be any sort of distance unit squared so again centimeters squared millimeters squared feet squared mile squared meter squared this means that we are taking a length unit and multiplying it against another length unit and because they both are going to share the same unit we end up squaring that to create the area the last dimensional measurement that we might take is a volume and a volume is going to be a length unit cubed so we have centimeters cubed millimeters cubed feet cubed miles cubed and meters cubed there are also units that are strictly used for volume such as liters pints and gallons there are some other ultrasound parameters that have units and those are going to include time which is usually measured in seconds milliseconds or microseconds we have velocity which is measured in centimeters per second or meters per second and frequency which is hertz kilohertz or megahertz now this is not an exhaustive list but these are some of the more common ones that you will see it's important to note that percentages technically aren't units but it is an acceptable numerical definition as it does refer to parts of a whole and the whole is 100 percent so if we say that something is 50 we know that is half of everything while it's not technically a unit we do consider percentages as an okay definition of a number now in america and maybe a couple other countries around the world we use what we call the empirical system so that means that we are describing distances in terms like feet or miles and our volumes are pints and gallons and we do weight and pounds and ounces there actually really isn't a whole lot of rhyme or reason to these distances but that's what we go with there is another system called the metric system and the metric system is used in the medical world and quite honestly by like 99 of the rest of the world because it just makes sense the metric system is based on tens and it's very easy to say that we have a meter which is a base unit the next unit up is 10 times the size of a meter and it's the next one is 10 times that and keeps going up by tens so the metric system has defined their base units and then uses prefixes to change the value of the base unit so some base units that we have in the metric system are meter for distance liter describes volumes grams describe mass seconds are usually used for time and hertz will describe frequency and this chart gives us a really nice overview of the metric system now this by no means is all of it but these are the ones that are going to be most applicable to ultrasound physics i want to tell you this is a memorized moment you're going to need to know the metric system and these values symbols and what it all means and how it all works together so let's go ahead and take a look at each section so let's go ahead and look at the prefix column first now as i mentioned earlier the prefix will change the value of the base so we often add the prefix to a base unit for example we have milliseconds centimeters kilograms megahertz you can see that we take a base unit and add a prefix to it to change its value above the base are large number so we have deca hecto kilo mega and giga and then below the base are small numbers now these are not negative numbers but these are values under one but greater than zero so we have deci centi milli micro and nano next we need to talk about what the values are of these prefixes the prefixes tell us that something is 10 times larger or 10 times smaller than the previous prefix so a meter can exist and then a decameter is 10 times bigger than a meter where a hectometer is ten times bigger than a deca meter but a hundred times bigger than the base kilometer is ten times bigger than a hecta meter but it is a thousand times bigger than the base so you can see that again we are basing everything off of tens as we get further away from the base working towards giga we are getting into larger numbers if we look on the other side of things though let's look at a base of seconds if we have a deci second that means it is 1 10 the time of a second a centi-second would be 1 100 of a second a millisecond is 1 1000 of a second so again we are working through tens going the other direction into very small numbers the next column gives us symbols so when we are writing out our units we can use the shorthand variation so for example if we were writing kilometers it would be a small k followed by a small m i do want to point out on the symbol column that for micro we have kind of a funny looking u it's actually a lowercase mu mu lowercase mu represents micro when you are typing things out if you don't have the capability of adding the greek letter mu you can just do a u it will understand what you mean or you can always just write out the whole thing micro whatever base unit capital m for mega but we've got a small m for milli and then you'll want to be familiar with your base abbreviations as well lastly i've included the exponents for these values 10 to the 0 means 1. so that goes along with the base there's no change to the base value but 10 to the sixth means to multiply something by a million ten to the sixth is one million so that matches up with the mega and the million value the exponents on this chart to help you later when we talk a little bit more about powers of tens but it's still a good idea to commit this to memory about which exponents go along with which values and also note that negative exponents again go with small numbers not negative numbers and lastly with the metric system you'll notice that some of our formulas do include multiple types of units we want to make sure that we are using complementary units when applicable so for example we talk about frequency in megahertz we need to talk about periods in microseconds because those are the complementary prefixes so giga and nano both mean billion and billionth mega and micro million and millionth kilo and milli are thousand and thousandth hecto and sendi are hundred and hundredth and deca and deci are ten and 10. so not only are you going to need to know what the prefixes are what the values are and what the exponents are you're also going to need to know how to convert between those units and as far as ultrasound goes we don't do a whole lot of conversion much more than like millimeters to centimeters maybe meters per second to centimeters per second but physics wants to make sure that you understand how this works so there will be a fair amount of converting in your tests and in your homework but before we get into converting metric system let's just go over what it means to convert units so it might be easier to first think about something smaller something more of a more tangible concept for us so let's think of four quarters we know that four quarters is the same as one dollar we have the quarter unit and we have the dollar unit four quarters is the same as one dollar we have not changed the value of the money that we have in our hands but we have changed the way we describe that value let's take a look at another example we have 365 days which is the same as a year 365 days is the same time frame as one year again we are not changing the value of the time that has gone by we are just changing how we describe it now unit conversion can be handy because it helps you to convert a number into more appropriate units for what you're talking about so for example we can say that ultrasound frequency is 17 million hertz there's absolutely nothing wrong with that we can also say then that the flu virus size is 0.0000 meters now nothing is wrong with that that is accurate but to better associate a unit with these very large and very small numbers it's better to convert them so 17 million hertz is the same as 17 megahertz where point zero zero zero one meters is the same as ten micrometers and it really just helps to put the concept a little bit more in context for the measurement that you are creating which brings us to the metric staircase i really like to teach the metric staircase as a tool for learning how to convert numbers now yes you can do the math and you can multiply and divide by factors of ten and thousand and all of that but there's also a really relatively easy way to kind of visualize it and to work through it if you're not proficient at doing the multiplication and division so let's go ahead and head over to the board and i'm going to go over some of these rules with you and then also go over some examples so first things first when you are writing out your metric staircase make sure that you're starting up at the upper left hand corner and working your way to the lower right hand corner when we go to write our staircase make sure that you are including all of your prefixes plus the blank stairs so we're going to start with giga and then we have two blank stairs mega two blank stairs kilo hecto and deca and then we get to our base and we're going to start writing our below the base prefixes so we have deci centi milli two blank stairs micro two blank stairs and nano so we should have nine stairs above and nine stairs below our base next rule that we need to know is that when we are converting down the staircase we are going to be moving our decimal to the right and we've got that hint down here we've got our right hand corner so we are going to move the decimal to the right when we are converting from a large number to a small number when we are converting up the staircase we are converting a small number to a large number and therefore need to move the decimal place to the left so let's begin by trying to convert a large number to a small number so let's go ahead and start with 3534 let's say hectoleters and we want to know how many hectoleters that is in milliliters so the first thing that you want to do is find your prefix that you're starting with so that's hecto and we're converting to milli the next thing that you're going to do is count the number of stairs so one two three four five we have gone down the staircase to the right five stairs that means that we need to move the decimal five places to the right so i'm going to rewrite the number just for clarity we've got 35 34 now typically we don't include the decimal place at the end we just leave it off because we understand that there are invisible zeros behind it but we're going to put it in for our visual cue to move the decimal place and again we're going to move it five places to the right so we're gonna go one two three four five and now our decimal place has moved to the end here five places but we need to fill in these gaps with zeros once you put the zeros in i like to kind of go through and just put my commas in it just helps me to better visualize what i'm looking at but i'm going to rewrite the number for clarity so in the end we had 3534 hectoleters converting into 353 million 400 000 milliliters so let's go ahead and try converting a number up the staircase going from a small number to a big number so let's say that we have 1230 milliseconds and we want to convert that into kiloseconds so again we're going to find our starting prefix and we're going to find our end prefix and then we're going to count to that stair so we have one two three four five six this time we have moved up the staircase six spaces so that tells us that we need to move the decimal point six places to the left so again i'm going to write our number and include our decimal point and then we're going to move that decimal point those six spaces so we're gonna go one two three four five six our decimal place moves here we can fill in our gaps with zeros we converted 1236 milliseconds into point zero zero one two three six kilo seconds and don't forget your units it's wrong until you put a unit on so let's go ahead and try two more examples they're actually going to be two examples from your metric staircase practice that is coming up we're going to look at the first one and the third one so again always practice writing out your staircase we have giga two spaces mega two spaces kilo hecto deca then we have base and then we have deci centi milli two stairs micro two stairs nano now the example on our metric staircase practice the first one wants us to take 0.1509 meters and convert that into nanometers so we are starting at a base this time and we are converting down to nanometers go ahead and count the stairs one two three four five six seven eight nine we are going to the right so we are going to move the decimal to the right nine spaces let's go ahead and write our number again and we'll get ready to move that decimal 1 2 3 4 5 6 seven eight nine our decimal moves to the end we fill in our gaps with zeros we can go back through and put in some commas if it helps us to visualize and in the end 0.1509 meters translates into 150 million 900 000 nanometers let's go ahead and take a look then at the third example get rid of some of this so i'm going to do the third example in a different color here so we're starting out with 15 million 820 000 nanohertz and we are going to convert that into hertz so we are starting this time at nanohertz and we are converting into a base unit a hertz so we are going to count our stairs going up so we have one two three four five six seven eight nine we have moved up the staircase nine stairs to the left is where our decimal needs to go to convert from a small number to a big number so let's go ahead and write that number out again here with our decimal at the end and we're gonna get ready to move it nine spots to the left one two three four five six seven eight nine we just have one gap to fill in here and now our new number is point zero one five eight two and the rest of the zeros are actually kind of not important for this but you can write them in if you want to and don't forget your units as i mentioned we did a couple of your practice problems already go ahead and pause the video finish up your practice problems and then come on back when you're ready to double check those answers and here they are so i'm not going to go through all of these with you but i will tell you some of the bigger mistakes that people make are either not writing out their staircase correctly missing those blank stares or not quite counting how to move the decimal places in the right direction or just missing filling in a zero or just kind of silly simple mistakes like that as i mentioned earlier a lot of the conversions that we are going to be doing are very simple millimeters to centimeters type conversions but the physics boards are going to want to know that you know how to convert if you feel like you want more practice with conversion there is a link at the bottom of your workbook page that you can click on and when you click on this link it will bring you to a google document when that google document opens up it's going to look like a bunch of mumbo-jumbo coding stuff and that's basically what it is there is a download button in the top corner you're going to want to click that download button and then go ahead and open it and when you open up that downloaded file if you open it up with some sort of internet browser such as chrome or edge it'll actually look like words and it will give you 20 metric conversions that you are able to do so it'll give you 20 problems and then if you scroll down on the page it'll give you all of the answers every time you refresh this page you will get new conversions so if you want a lot of practice with this this is never ending practice it is all randomly generated for you to work with so now that your metric system masters i want to revisit the chart because i do want to talk a little bit more about exponents that match up with the prefixes so when we have a base 10 and raise it to an exponent we talk about it being the powers of 10. you'll note that giga means billion and 10 to the ninth also means billion and that is why these two go together note that numbers that are above the base have positive exponents and the numbers get large where the exponents below the base are negative and remember that does not mean that they are negative numbers but rather make very tiny numbers like 1 billionth but what does it mean to raise 10 to a power or create an exponent well let's use 10 to the ninth as an example now to raise ten to the ninth means to multiply one by ten nine times so we take one times ten equals 10 10 times 10 equals 100 times 10 is a thousand times 10 is 10 000 and so on all the way until we've done that nine times and get to one billion so when raising a number to a positive power the number is going to get larger let's take a look down then at nano where we have 10 to the negative ninth power when we raise 10 to the negative ninth power that means to divide 1 by ten nine times over so we have one divided by ten equals one tenth divide that by ten we get one one hundredth and divide that by ten and we get one one thousandth and that's going to keep going until we get to one billionth so when we raise a number to a negative power the number gets very small i had mentioned that powers of 10 are associated with scientific notation i want to go over a couple more rules and then we're going to head over to the board so i can visually show you what i'm talking about when we convert to scientific notation and convert out of scientific notation so let's first take a look at example with a billion so we can take one billion we can write it out with all the zeros that's a really pretty big number we've got a lot of zeros back there we can convert that into scientific notation by writing it as one times ten to the ninth if we start with one times ten to the ninth and we want to convert that back out of scientific notation then we need to take a look at what the power is recognize if it's positive or negative and it's positive in this case and that will tell us to move the decimal to the right to expand out of scientific notation we're going to look at the rules done for negative powers if we start with 1 billionth that's a whole lot of 0.0001 that's the same as saying 1 times 10 to the negative 9th when we want to expand scientific notation out with a negative power then we need to move our decimal to the left and just a few more rules that i want to go over with you before i start to show you how this works on paper so if the expanded number is greater than 10 then we should have a positive power of 10. if the expanded number is between 1 and 10 then the power of 10 is 0 and if the expanded number is less than 1 but more than 0 then our power of 10 is negative so when any number is raised to the power of 0 it's going to equal 1. let's head over to the board so i can show you what i'm talking about with converting into scientific notation and expanding out of scientific notation let's take a look at the first example that i gave you with one billion we can write 1 billion out with all of our zeros and let's say we want to convert this into scientific notation well the first thing that you need to do with a large number is place a decimal point so you have a number that is between 1 and 10. so if we were to place the decimal point here that actually is 1 million that's not a number between 1 and 10. if we placed it here we would have the number 100 that's not a number between 1 and 10. so typically you're going to place it towards the beginning of the number and here is where we have the value of one and that is between the values of one and ten the next thing you wanna do then is count the number of spaces that sit behind the decimal so we have one two three four five six seven eight nine so that tells us that we are taking one and multiplying it by ten to the ninth let's take a look at another example this time we'll use a number let's say we'll use 2 355 000 as our example and we want to convert that into scientific notation well again we have to find a number that's between 1 and 10 and place our value there so if i placed my decimal here that doesn't work because now i've got a value of two thousand three hundred fifty-five that's not between one and ten so we actually need our decimal point to be here so that gives us a number of two point three five five and now we need to count the spaces behind the decimal so one two three four five six so two point three five five multiplied by ten to the six is the same as two million three hundred fifty-five thousand and if you recall from our chart we know that ten to the sixth lines up with million let's take a look then at what it means to take a very small number write it into scientific notation so let's use our example again of 1 billionth and this time what we're going to do is we're going to start with moving our decimal and we want to move our decimal to the place that gives us a new value between 1 and 10. so here doesn't give us that value here doesn't and it's actually all the way down here so now we have a value of 1 which is between 1 and 10. and all we need to do is count how many spaces we moved that decimal point so we have 1 2 3 4 5 6 7 8 9 moves and because this is a very small number we want to make sure that we are using a negative power so 10 to the negative ninth let's try an example with a number other than one again let's say this time we have point zero zero zero three three one and we want to convert this into scientific notation so again the first thing we need to do is figure out where do we move the decimal place to make a number that is somewhere between 1 and 10. we have 1 2 3 4 and here we have 3.31 if we leave it there so we move that four spots over we're making a small number so we need a negative so three point three one times ten to the negative fourth is the same as point zero zero three three one now let's take an example of what it means to expand scientific notation let's say that we start with the value 3.4 multiplied by 10 to the fifth the first things we're going to do are look at what the power is and this is a positive power so this means that we are going to be getting a bigger number than 10. we can take our three point four and the positive power tells us to move our decimal five spots to the right so we're gonna go one two three four five and fill in our zeros so three point four times ten to the fifth is the same as three hundred and forty thousand now another way that you can do this if you are inclined you can take that 3.4 and you can multiply it by 10 to the fifth or the expanded version of 10 to the fifth which would be 100 000 and that would also give us the answer of 340 000. so it's up to you how you want to do it if it's easier for you to count spaces then by all means do what is easiest what you understand let's go ahead and look at a number that needs expanding with a negative power so this time let's do 8.71 multiplied by 10 to the negative seventh so we're going to write our number down and we're going to look at what the power is the power is a negative so that means we need to move the decimal to the left seven spots so we are going to move one two three four five six seven fill in our gaps with zeros and eight point seven one times ten to the negative seventh is point zero zero zero zero zero zero eight seven one now just to make sure that we are rounding out all of the information and rules that i gave you let's take a look at one more example let's say that we have a number that is already between 1 and 10. so let's say our number is 5.4 and somebody says write that into scientific notation well we can't move the decimal anywhere to make it into a number that's in between the value of 1 and 10 so because we move the decimal zero spaces we are multiplying 5.4 times 10 to the zero and that is going to work out because ten to the zero equals one so we are essentially saying five point four times one and in reality any number raised to a zero is going to equal one so two to the zero equals one three to the zero equals one same is going to be true then if you need to expand a value that is already in scientific notation and it's multiplied by 10 to the zero you're going to automatically know that that is one and that this number is already expanded go ahead and pause the video you have seven problems that you can try expanding and writing into scientific notation go ahead and unpause the video when you're ready to check out those answers and here they are so make sure that you are double checking your answers make sure that you understand the concept of going larger than 10 is a positive power less than 1 is a negative power and when the power equals 0 it does not change the value of the original number because 10 to the 0 equals one so now that you know about powers of ten let's talk about powers of other numbers we just learned that ten times two is the same as saying one times ten times ten which equals one hundred we took one multiplied it by 10 twice well what if we change the base number to the number two and we do two squared well that's the same as multiplying one times two times two which equals four we took the value 1 multiplied it by 2 twice because that's what the exponent told us to do what if we raised the exponent to 4 though well now we have 1 multiplied by 2 times 2 times 2 times 2 which equals 16. this means that that was 1 multiplied by 2 4 times or to the exponent numbers one more example here we have 3 as our base if we square 3 that means to multiply one by three by three which equals nine so that is another way of saying multiply one by three two times so really the generic way of thinking about powers and numbers is to take the base number and look at the power you're then going to multiply one times the base number however many times the power tells you to do it now there are some special ways of saying to the second exponent or raised to the second power when we raise to a power of 2 it means to square something and when we raise to a power of 3 it means to cube it as far as knowing how to raise other numbers to powers what i suggest that you focus on or make sure that you understand is to do basic squares so 2 squared 4 squared 5 squared so on and then also look at raising the base number two to exponents zero through eight so you should know two to the zero is one where two to the eight is two hundred and fifty six those numbers are going to be very helpful for you on your physics boards now all those exponents that we just talked about were all positive exponents what if we raised to a negative exponent well we know that raising 10 to a negative power meant to divide 1 by 10 that many times over well the same concept is going to be true for other numbers but it might be a little bit easier to think of it this way when we have a base number and raise that to a negative exponent it's the same thing as saying 1 divided by that base raised to a positive exponent so remember 10 to the negative 2 was .001 or 1 100th we learned that 2 squared was 4 but 2 to the negative 2 equals 1 4. again we learned 2 to the fourth was 16 but 2 to the negative 4 is 1 16 and 3 squared was 9 but 3 to the negative 2 is 1 9. hopefully this little trick will help you to not get distracted by the negative power now on the last slide i just showed you that if you take 1 and divide it by a base number raised to a positive power you will figure out what that base number is raised to the same negative power and what we ended up coming up with were a bunch of fractions but what happens when fractions aren't an answer well you need to know how to convert a fraction into a decimal and here's how we do it we're going to do it with long division recognizing what the numerator is and what the denominator is let's head over to the board and i'll show you how this works out so let's say you're taking your test and one of the questions asks you what is 2 to the negative 2 and being the really smart student that you are you know that you can ignore the negative and you know 2 squared is 4 but because of the negative power we just have to put it all under a one so we know that two negative squared is one fourth but what if one fourth isn't an answer that's available to you what if they're all decimal numbers we need to be able to convert fractions into decimals and we do that by understanding the numerator and the denominator and using long division so to convert 1 4 into a decimal we are going to put our long division symbol down we are going to put the numerator inside the long division and we're going to put the denominator outside of the long division we want to make sure that we include a decimal because we are using fractions we're going to be numbers less than one so we want to put our decimal here and we want to carry the decimal up above the long division bar now the first thing we're going to look at does 4 go into 1 well it doesn't so we actually need to add zero can four go into ten absolutely and it goes in twice we'll complete the long division two multiplied by four is eight ten minus eight is two now we're going to see if 4 can go into 2 and it cannot so we're going to add a 0 up here and drop it down 4 can go into 20 5 times so we're going to multiply 5 times 4 that equals 20 we subtract that from the original 20 and we get zero so we've actually completely converted one-fourth into an ending decimal of 0.25 let's go ahead and try another example this time let's try to convert 5 6 into a decimal again we have our numerator on top and our denominator on the bottom draw in our long division sign the numerator goes on the inside the denominator on the outside bring in your decimal point and raise it up six cannot go into five so we need our first zero six can go into fifty eight times six is forty eight subtract the two and we get two now six cannot go into two so we need another zero drop that zero down 6 goes into 20 3 times 3 times 6 is 18. subtract and we get 2. now you might start to see a pattern here and this is going to be a repeating number six cannot go into two so we'll need another zero drop it down six goes into twenty three times three times six is eighteen and we get two so you can see that we are constantly going to be getting the remainder of two and having to add in zeros which means that the number will just keep repeating so 5 6 is the same as 0.83 repeating now that we're back let's go ahead and pause the video and you can take some time to practice converting fractions in your workbook when you're ready unpause the video and come on back so here are your answers as you can see we did do one of your examples i wanted to show you what it looked like to have a repeating decimal i will say that you will not probably have to do this too often but it is a handy tool to have in your toolbox in the event that you need it now i mentioned reciprocals back when we were talking about relationships in formulas and that was because the reciprocal was a special relationship that meant when two variables were multiplied they would equal one so a times b equals one a and b are reciprocals of one another because the one is constant it's not going to change and it means that the variables or the factors in this formula are the items that are uniquely related it means if one goes up the other must go down but they do this in a really particular way so i want you to see if you can figure out what the pattern is from these examples of reciprocals so 5 once multiplied by one-fifth equals one thirty-four multiplied by one-thirty-fourth equals one and seven-sixteenths multiplied by sixteen-sevenths also equals 1. so maybe you've noticed that if you take a move it to the denominator position or flip it around you will get your reciprocal and that will equal one this pattern is the only way that you will be able to multiply two numbers together so that they will always equal one now there are two formulas that we actually use in physics and we'll learn about them in the next few chapters that are reciprocals and they are frequency times period equals one and pulse repetition frequency multiplied by pulse repetition period also equals one so the relationship of reciprocals is actually applicable to ultrasound physics now one last thing to kind of review what we just talked about with converting fractions one last thing that i want to touch on then is to remind you that you won't always have fractions as your answers so be prepared to convert fractions into decimals so you can understand what the answers are so 5 multiplied by 0.2 which is the same as 1 5 will equal 1. so make sure that you are comfortable converting fractions and recognizing all right we're to our last mathy thing that we need to talk about most of you are probably familiar with the idea of an x-axis and a y-axis if we take a look at a 2-d box we know the horizontal line is x and the vertical line is y but when we introduce a cube or a third dimension to these two axes we get the z axis and the z axis actually gives us a sense of 3d world a lot of the ultrasound information that we collect is based on mapping data onto a graph so for example i have a spectral tracing image here along the x-axis or the horizontal line we have time going by and on the vertical axis or the y-axis we are displaying velocity we definitely will be covering more about what is displayed on each axis but this serves as just a reminder that we do have the x y and z axis to represent the 3d world and that learners is the end of our first unit make sure that you are going through your workbook and going through your nerd check questions so you are prepared for your quiz