Mathematics and logic

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(Disclaimer: I have ADHD and am completely useless when it comes to math. So please forgive me for being super-stupid on the subject.)

To my understanding, Mathematics are seen as completely logical. Which I don’t have a problem with except for when it comes to one certain thing in math that I just can’t make sense of as being considered logical:

Rounding of decimals.

To my understanding, the rule is that when you have a decimal that is 5 or higher, you round up. If 4 or lower you round down.

Two things that I don’t understand about this:

1. When you round up, you magically pull value out of the air that wasn’t there to begin with, and do the opposite when rounding down. How is this considered logical?

2. The rule isn’t applied universally. I’ve seen cases when, for example, making store purchases, no matter how low the decimal, it is rounded up and not down.

I appreciate any help you guys can give. Thank you in advance for the assistance! <3

In: 6

10 Answers

Anonymous 0 Comments

It is an arbitrary convention based on good reasonong.

In a significant digit, this is the range of possible integer values:

0, 1 ,2, 3, 4, 5, 6, 7, 8, 9

Since each subsequent order of magnitude begins with a last significant digit of “0,” then it makes sense to divide the list above in half between 4 and 5.

Anonymous 0 Comments

You’re reciting the “rule” that people learned in elementary school. But, that’s not actually “the” rule. In fact, there isn’t a single rule. Sometimes people round up (common if, for example, you’re computing sales taxes) or round down (common if, for example, you’re paying out interest on a savings account).

Then, there’s “round to the nearest…. ,” but that leave some ambiguity — what do you do if you’re *exactly* half-way between? This is the problem our elementary school teachers were trying to solve by saying “if it’s 0.5, you round that up to 1 and not down to 0.” But, that’s just a convention that was useful in elementary school to ensure that every student got the same answer. But, in real life, sometimes, people will round 0.5 down to 0.

The good thing is that this actually doesn’t come up all that often — when you’re rounding, it’s usually because you’re doing a measurement of some sort. And, it’s really rare for measurements to be *exactly* half way. You almost never have “7.5 miles.” It’s always something like 7.4995 or 7.5001 miles.

In the few cases out there where it’s possible to be half way (“these nails are half a cent each”), always a good idea to ask how rounding happens.

Anonymous 0 Comments

“Rounding” is less mathematics than it is a useful convention for counting things. It’s not based in mathematical logic, but rather just rules we set up for counting things.

The rounding your describe is the type that we use in counting things when we want to make adding up numbers in our head easier, but still get close to the real number.

There’s another type of rounding where you simply take a value between two others and count it as the greater value, called “rounding up”. People do it for things like planning food for a party where they prefer to over-estimate how much to buy so that they don’t run out of food by buying too little.

Sometimes people do the opposite, “rounding down” by taking the lower number with the understanding that they are under-estimating. People do this when buying big boxes of stuff to make sure that they don’t buy so much that it goes to waste.

People also round off by irregular amounts, not just haves. If you cook and you need to change how much of a recipe you make, you multiple or divide the ingredient amounts in the recipe to adjust how much you make. The problem is that your spoons and measuring cups only come in so many sizes, so sometimes your just round off to the nearest whatever-size-scoop-thing-I-have of an ingredient. It applies anywhere that things come in fixed “chunks” or units of a given size and you can’t or don’t want to break it up.

Mathematics, however, can describe the error in using each method, the assumptions you make about the distribution of the things you are counting, etc.

Anonymous 0 Comments

Mathematics are built on logic. But so is physics, therefore chemistry, therefore biology, therefore psychology, and we’re not completely logical, are we?

Rounding isn’t a matter of math, it’s a matter of presentation. We write either π or 3.14, we don’t write every known digit of pi. Not because π=3.14, but because we have no use for so much precision, and because they wouldn’t fit meaningfully on a page.

As a method of presenting numbers, the intent of the presentation matters. Usually, you round to the “nearest” number because it balances an accurate representation of reality against the limitations of human perception.

If you’re buying a new airplane, you’ll round the price to the nearest couple digits of a million dollars on marketing materials, but you’ll round to the nearest cent on the bill. You’ll round down if the question is “how many whole apples have you got” because 3/4 of an apple isn’t a whole apple. You’ll round up to the nearest cent because the halfpenny is discontinued. All of this is done according to the interests of whomever presents that number. Of course a store will round up if they can get away with it.

Anonymous 0 Comments

Rounding isn’t really “math” math. It’s just convenience step you perform after you’ve done the computations, since for us humans it’s just easier to deal with rounded numbers.

Additionally, when rounding does happen, it’s supposed to shave off or add amounts that are too small to be relevant to the situation. When you’re talking about your weight, for example, going plus or minus two grams shouldn’t be relevant in any situation where your weight comes up.

Anonymous 0 Comments

Thank you everyone! I get it now! You’re all awesome! <3

Anonymous 0 Comments

I don’t think anyone’s quite addressed the question yet. There’s a couple of things going on.

First, rounding is *convention*. It doesn’t make sense to talk about it being “logical” or “not logical”. It’s something we collectively agree to do in certain situations, not something you can prove.

For instance, I can prove, beyond a shadow of doubt, that there are infinitely many prime numbers or that it’s impossible to construct a regular heptagon with a straightedge and compass. These are mathematical *theorems*. I can’t, on the other hand, “prove” that 0.63 rounded should be 1 instead of 0.6, because we haven’t given rounding clearly defined rules. This is by design: We use it in whichever way is most convenient at the time.

Note that if I instead talk about “rounding to the nearest tenth”, then *that’s* a rigorously defined operation and I can start making factual statements about it. I can say that rounding 0.63 to the nearest tenth is 0.6, not 0.7 or 1.

I can also say that “rounding 0.63 gives 0.6”, and ask you to infer from that statement that I *meant* “rounding to the nearest tenth”. If you then turn around and say “0.63 rounded is 1”, you’re not *wrong* for disagreeing with me, we’re just using ambiguous language in different ways.

Second, math is *abstract.* I encourage you to let go of the idea that numbers embody “value”, or that there was anything “there to begin with”. Rounding is just another operation that takes a number as an input and spits out another number as an output, no different from, say, adding 1, or squaring, or taking the cube root.

Anonymous 0 Comments

Rounding numbers is a practical thing, so technically you are allowed to do it however you wish. What you’ve described is one convention, written down in textbooks. What the stores do is another convention that’s probably written down somewhere in their policies. There’s also something called “banker’s rounding” which rounds halfs to the nearest even number, so 5.5 becomes 6 and 8.5 becomes 8. In short, there are a lot of various rounding methods used for different purposes.

Anonymous 0 Comments

I love your view of it 😀

You don’t magically add value… You approximate.. we do that when for the sake of simplicity we can lose precision

It’s like saying “bus arrives in hour” instead of “bus arrives in 58 minutes and 47 seconds” …

It is not logical…. It’s just simplified

And when it comes to stores… That’s not logical math, that’s economic and capitalism…. They round it up not because it is correct, but because they make more

Anonymous 0 Comments

Rounding sacrifices precision for convenience. Essentially, there are plenty of situations where you don’t want to have a long string of numbers after the decimal point so you just pick a point to cut it off. We use the .5 as a rule of thumb in most cases just so everyone is on the same page, but in some cases it makes sense to always round up or down.
For example, when I’m doing carpentry, maybe I’ll need a piece of wood that’s between two of the marks on my tape measure. In that case I’ll round up for two reasons. I don’t have the equipment to make a precise enough measurement so I have to round. And I’m always going to round up because it’s easy to shave off a little more later if I need to.
Rounding is all about making numbers easier to use. So the logic comes in to determine how to best round a number to fit the situation.