Microscopy magnification problem

[FancySnow]

Hello,

I'm having trouble understanding how and why the math is the way that it is to get the answer. The question states:

A specimen has a diameter of 1.5 micrometer. What is the minimum magnification that will allow a human to resolve this object?

150 micrometer/1.5 micrometer = 100X <-- that is the answer

My question is: why do I have to do a division? I don't understand the why I have to do. I know how to do it, but why? How? How does micrometer/micrometer = minimum magnification? Can someone explain? The book says that the resolution of the human retina is about 150 micrometer. Please enlighten me.

Thanks!

 

[Drakkith]

Rearrange the equation to yield $100 * 1.5 μm = 150 μm$.
This equation means that your current 1.5 micrometer specimen needs to be multiplied by 100 to get up to the minimum 150 micrometer required for viewing by the eye. So the magnification is 100x, since that's what magnification means mathematically.

Think of this rearranged equation as your fundamental equation linking the specimen size, the magnification, and the resolution of the eye. If you are missing one of the three variables, then you just have to use standard algebra techniques to solve for it. In this case we wanted the magnification and we already had the other two variables. Rearranging the equation and dividing is nothing more than using algebra to solve for our missing variable. It's not that there is some mystical something or another that makes micrometer/micrometer = magnification, it's just that our fundamental equation is set up in such a way as to require that division if you're solving for magnification.

Also, note that magnification is unit-less, which is exactly what happens if you divide micrometers by micrometers. The micrometers cancel out and you're left with no units. So there's nothing mystical about it, it's just math.

 

[FancySnow]

Rearrange the equation to yield $100 * 1.5 μm = 150 μm$.
This equation means that your current 1.5 micrometer specimen needs to be multiplied by 100 to get up to the minimum 150 micrometer required for viewing by the eye. So the magnification is 100x, since that's what magnification means mathematically.

Think of this rearranged equation as your fundamental equation linking the specimen size, the magnification, and the resolution of the eye. If you are missing one of the three variables, then you just have to use standard algebra techniques to solve for it. In this case we wanted the magnification and we already had the other two variables. Rearranging the equation and dividing is nothing more than using algebra to solve for our missing variable. It's not that there is some mystical something or another that makes micrometer/micrometer = magnification, it's just that our fundamental equation is set up in such a way as to require that division if you're solving for magnification.

Also, note that magnification is unit-less, which is exactly what happens if you divide micrometers by micrometers. The micrometers cancel out and you're left with no units. So there's nothing mystical about it, it's just math.

Thanks! But, to be honest, there seems to still be a disconnect. I am not really understanding why the math is done the way that it is done to get magnification. Thanks for clarifying that magnification is unit-less - that make sense. But, I'm honestly still stuck on the rationale say...

150 micrometer/1.5 micrometer = magnification. Even if magnification is unit-less. I don't know why the dots are not connecting.

. Other type of numbers are unit-less too right? I feel like I'm missing a piece of information that would truly elucidate this for me.

 

[Drakkith]

But, to be honest, there seems to still be a disconnect. I am not really understanding why the math is done the way that it is done to get magnification. Thanks for clarifying that magnification is unit-less - that make sense. But, I'm honestly still stuck on the rationale say...

It's nothing more than the fact that you need to increase the dimensions of the sample by 100x to bring it up to the minimum size to be seen. If you take all the dimensions, meaning the length, width, height, radius, etc, and you multiply them by 100, then your sample is now large enough to be seen. That's it. There's nothing more to it than that.

Magnification literally just refers to the amount that the dimensions are multiplied by. A pair of binoculars with a magnification of 5x means that the image appears to have had all its dimensions multiplied by five. Hence it looks 5-times larger. Notice that the 'x' in 5x and 'times' in 5-times both refer to multiplication.

150 micrometer/1.5 micrometer = magnification. Even if magnification is unit-less. I don't know why the dots are not connecting.

. Other type of numbers are unit-less too right? I feel like I'm missing a piece of information that would truly elucidate this for me.

The fact that it is unit-less is not particularly significant. I only brought it up to show you that the division of micrometers by micrometers gives you a unit-less number, which is exactly what you need if you want magnification since magnification is unit-less as well. They key thing to understand is that we need to multiply the dimensions of our sample by some value if we want to make it visible. The amount that this value takes is our magnification. That's it. Then when we want to solve for the magnification we have to divide because of the rules of algebra. If magnification and sample size are multiplied together on one side of the equation, then we have to divide out the sample size from both sides in order to solve for magnification.