Applying the rules of significant figures to calculations

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Applying the rules of significant figures is important in fields like chemistry and physics, where accuracy is crucial. While everyday calculations may not require strict adherence to significant figures, they help convey the precision of measurements. In engineering, physics, and math, significant figures become essential, especially when dealing with equipment limitations. Calculations should be carried out fully before rounding to the appropriate number of significant figures based on the precision of the tools used. Ultimately, understanding and applying significant figures enhances the accuracy of scientific calculations.
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I have a question about applying the rules of significant figures to calculations. Should one apply the rules to everyday calculations? I don't know if it will benefit me in the long run to count the number of sig/decimal places in numbers in the simplest calculations. If I major in engineering/physics/math, will I need to use sig figures everyday of my life or what? Do you need to use significant figures in chemistry? physics? everything related to math and calculations? thanks.
 
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In mathematics it's ideal to get exact answers, and you should be.

Significant figures are useful in real life for chemistry and physics. But you carry out all your calculations until the very end. Then use your significant figures.

It's just telling you your accuracy, if you have a meter stick and tried to measure to the micrometer. It's not going to happen, it's purely guessing up to a certain point and anything after won't matter.

If you had a scale that only read to 2 decimal places, you can get a number with decimal places after calculations. But you're only as accurate as your equipment, so you'd count to two decimal places in the end.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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