### Video Transcript

In this video, we’re going to learn about SI unit multiples and submultiples. What they are. Why they’re important. And how to use them in example problems. To start off this topic, let’s first think about the importance of having a consistent set of units to work from. Imagine that we meet an architect, someone who designs homes. The architect is very excited to show us plans for a new house that she’s just designed. And let’s further imagine that when designing the house, the architect used a unit of distance called the splot. So one splot or two splots to represent different distances in the dimensions of the designed house.

Now imagine that we come on scene. And our unit of measurement that we’re used to is not the splot, where we use a unit called a grang to measure distance rather than a splot. And clearly, one grang is not equal to one splot. Say that as we’re discussing the plans for the house, we ask the architect how tall it is. The architect responds: 13.8 splots. Now if we haven’t heard of splots and certainly don’t use it as a unit of measurement, this will be a very confusing answer to us. We might go and try to find out what one splot is, compared to a grang. And express the height of the house in a unit that made more sense and was more familiar to us. Now even if we did convert the height of the house into units the architect was more used to. Ideally, we’d both be able to agree on a common set of units to discuss the plans in. Having a common set of units to refer to will help us out whether we’re talking about units of distance or units of time or units of mass or really any measured quantity.

It was with this goal in mind that the SI or System International set of units were developed. We see from the name that the intention is that this set of units be useful regardless of where we come from. The SI system has seven of what are called base units. The first base unit, the meter, measures length. The second measures time. The mole measures the amount of a substance. The ampere measures electric current. Kelvin measures temperature. The unit candela measures luminous intensity or brightness of a light source. And lastly, the kilogram is the unit that measures mass. Nearly any quantity that we would want to measure can be expressed in terms of these seven base units. For example, when we measure force, which is in units of newtons, that unit of newtons can be expressed in terms of the base units kilograms, meters, and seconds.

So we’re at the point now where we have a common language for describing measured quantities. But we can go a step further in terms of being able to describe those measured numbers. Imagine that you purchase a green laser pointer. And as you open it out of the box and start trying the pointer out, you notice some writing on the side of the pointer. Looking more closely at the writing, you see it says. Wavelength 𝜆 equals 5.50 times 10 to the negative seventh meters. You recognize that unit, meters. It’s one of the base units in the SI system. And you recognize, roughly, how long a meter is. But the number in front of that unit, 5.50 times 10 to the negative seven, is a little bit hard to understand. It’s apparently very small. But it’s hard to get a feel for what that number means. This is where an extension, built on top of the SI system of base units, can help us. That is, we’ll be helped to understand what this wavelength number means by referring to SI unit multiples and submultiples. Let’s look at a table of those multiples and submultiples now.

In this table on the left-hand side, we have multiples of SI units. And on the right-hand side, we have submultiples. Let’s look on the multiple side of the prefixes and their meanings. All the prefixes that we see apply to every single one of the seven SI base units. For example, we could have a hectometer. That would be 10 to the two or 100 meters. Or, we could have a gigasecond. That would be 10 to the ninth or a billion seconds. Or, we could have a peta-ampere, 10 to the 15th or a trillion thousand amperes. The purpose of these multiples is to help us be able to express and easily understand large measured values. In a similar way, the submultiples are designed to help us be able to easily express and understand small measured values.

If we wanted to measure a very short distance, we might measure it in millimeters or 10 to the negative third meters. Very fast optical processes involve times measured in femtoseconds. That is, one thousandth of a trillionth of a second. And if you recall back to our green laser pointer, the value of the wavelength of that pointer might be better expressed in terms of a nanometer. That is, a billionth of a meter. It’s not necessary to have all the prefixes and their meanings memorized. But it is helpful to have a few in mind as we work through different exercises. Let’s get some practice using multiples and submultiples in a couple exercises now.

25000000 meters squared equals blank kilometers squared.

We want to solve for the value that goes in the blank. And to do that, we can recall that one meter, the SI base unit of distance, is equal to 10 to the negative third kilometers. So if we want to convert 25000000 meters squared into kilometers squared, we would multiply our original given number by that conversion factor two times over. Because we have a squared value, to express this number in kilometers squared. When we multiply these three numbers together, we find they equal 25. So 25 million meters squared is equal to 25 kilometers squared. Now let’s try a second exercise with multiples and submultiples.

A device called an insolation meter is used to measure the intensity of sunlight. It has an area of 100 centimeters squared and registers 6.50 watts. What is the intensity in watts per meters squared?

We’re given an area of 100 centimeters squared and a power rating of 6.50 watts. We want to express the intensity of this radiation in units of watts per meters squared. Though we’ve been given them in units of watts per centimeters squared. In terms of the given values, we can write that 𝐼, the intensity, is equal to 6.50 watts divided by 100 centimeters squared. To convert our denominator into units of meters squared, we can recall that one centimeter is equal to 10 to the negative two times our SI base unit of meters. That means if we take 100, the number of square centimeters given, and multiply it by 10 to the negative two two times. Then the product of those three numbers will equal a value in meters squared, the unit we’re interested in. When we multiply these three numbers together, we find that 100 centimeters squared is ~~1.01~~ [0.01] meters squared. That means we can rewrite our intensity as 6.50 watts divided by ~~1.01~~ [0.01] meters squared. And when we calculate this fraction, we find it’s equal to 650 watts per meters squared. That’s the intensity of the sunlight measured in these units.

In summary of this topic, SI unit multiples and submultiples make number values easier to understand. It’s important to realize they don’t change the actual measured values. They just change the way those values are expressed. And we use them particularly with very large or very small numbers. That’s where their utility really comes through.