math is hard

Intuitively speaking, how many times does half of a thing fit into a quarter of a thing? The answer is, exactly one half time.

How many halves fit into a quarter? Half of them

That's exactly how I think of it, so strange someone downvoted you

✅ Math is hard

❌ This math is hard

0.25 / 0.5 = 0.5
0.25 = 0.5 × 0.5
1/4 = 1/2 × 1/2

p/q=q

So q=√p

Works with a lot of numbers ☝🏻🤓

Ehhh |q| = √p but close enough
- If q=√p then -q=√p also.

This is why "divide by half" and "divide in half" are two different things

It won’t keep you up if you just think of Divide as just multiplying by the fraction

Yeah I was gonna comment that 0.25*(1/0.5) = 0.5 doesn't look nearly as weird
- I didn’t specify fully, but I was just thinking 1/4 * 2/1
The math looks perfectly fine. But when people phrase "half of a quarter" I think they have (1/2)*(1/4) in mind, instead of 0.25/0.5
- But this isn't "half of a quarter" this is "the reciprocal of a half, of a quarter"
  Half of a quarter is 0.25/2 or 0.25*1/2

If you give half a person a quarter of a thing, how much would you be giving a full person? That's right baby, half a thing. Don't sweat it.

I worry that people who can get onto the internet find fractions a challenge.

I don't think this was shared because people are finding it a "challenge" it just looks funny.
It takes all of a few seconds for your actual mathematical processing to kick in and you go "oh yeah duh" but its just a funky little string of numbers.
It lives in the same camp as how none of the >3 whole multiples of 17 feel like multiples of 17. 68? Preposterous.
It's why my favorite way to troll the usual "why isn't everyone on metric" goombahs is to tell them they're just too lazy and/or dumb to do math with fractions.
- I mean, I'm too lazy, but really - why make life harder than it has to be?

x / sqrt(x) = sqrt(x)

Damn who would've thought?

Divide by 1/2 or multiply with 2/1. It's an equivalent transformation.

I just think of division as how many times the right expression fits inside the left expression. 0.5 fits into 0.25 only 0.5 aka 1/2 times, because only half of it fits.

Precisely this. The people not getting the OP are why Common Core was developed.

The numbers between zero and one are where all of the fun is!

1×2=2

Wow. Much brain. Maths wow.

A quarter is one half of one half. Makes perfect sense.

That would be 0.25 = 0.5*0.5 and rearranged equation doesn't help unless you understand it anyway

25/5=5

2.5/0.5=5

o.O

I think, it is the real world logic that makes it hard to grasp. If you divide something with something small it becomes bigger. Mathematically it's easy and makes sense, but it it's somehow not intuitive. Especially for young me :)
That's the same as 2/2=1 3/3=1 268/268=1 ...
Except that's not what's written.
It's 0.25/0.5
Raising it exponentially would be
2.5/5, or
25/50=50
1/2≠50. Does not compute.

This just comes down to the fact that "dividing by a fraction is the same as multiplying by the inverse of the fraction" is an easy rule to follow but not particularly intuitive. In natural language, when most people hear "divide by half" they're actually picturing "divide by two" in their head.

This don't avoid to sleep not even for 1/2 second. But pick any number. If that number is even, divide it by 2. If it's odd, multiply it by 3 and add 1. Now repeat the process with your new number. If you keep going, you'll eventually end up at 1.

You can't prove that!
- Yes, but everyone tried since a century to find a number with which it don't work, good to avoid sleep.
https://www.smbc-comics.com/comic/math-translations
Any positive number?

It’s going to be okay:

Edited to add this: Singapore math insists however, that we eliminate the use of visuals in describing arithmetic within the rationals. They encourage that users of common core rely upon the number line, and solely the number line for thorough and most mathematically sound representations of arithmetic, even when involving the division of fractions.

For those not up to speed to with common core, remember how the teacher used to draw a diagram of a bunny hopping from one integer to the next integer to represent adding given integers, such as 4+3, or -2+1? Imagine that representation being used with problems like 1/7 divided by 5/49, and no decimal approximation is allowed. It’s fascinating and truly something to appreciate from the standpoint of someone who truly loves mathematics. I think it makes for great discussions amongst math graduates like myself, and other math enthusiasts. What does that mean for those who are not so enthused? Sometimes it means the teacher receives death threats from angry students. You can’t make everyone happy.

I’m not sure I completely agree with the number-line-only approach, but I’m definitely sympathetic to it. It reinforces the idea that fractions are numbers like any other numbers, and not pieces of pizza.
- I get that. I like the number line approach, and respect it, but I have also observed seasoned math coaches fumble the visual explanation of a division by fractions problem where the numerators and denominations were relatively prime. As soon as the guy had drawn the first fraction and began to say, “we’d multiply by the recipro-…”, I could tell it was going to be long problem. He just stood there, and then asked, “well, how would I go about explaining the ‘keep change flip’, if you will?” He ended the problem by saying he might just explain that the distance drawn for the first fraction needs to be repeated on the other side of the fraction to show the multiplication by the denominator of the second fraction, and then that distance could be broken into parts to demonstrate the division by the previous numerator of the second fraction.
  Basically he ended the problem by saying, “let’s just reflect it! Then we can break it up.” There wasn’t really a sound justification for the reflection piece of the process, other than saying, “we need to multiply by the reciprocal of the second fraction, so we’ll just have to multiply by its denominator it had, prior to flipping it.”
  That was the quietest meeting I have ever seen amongst that group of adults.

Multiplication of x times 6:

x * 6 = 1/2 x * 10 + x

This can sometimes be a shortcut for numbers that are easier to divide by 2 than to multiply by 6.

Take half as tens and add the number.

6 * 6 = 30 + 6 = 36

8 * 6 = 40 + 8 = 48

150 * 6 = 750 + 150 = 900

320 = 1600 + 320 = 1920

Etc.

Sleep well.

So an extension of the x * 5 = x/2 * 10 shortcut
- Exactly. Multiples of 5 are easy enough in my opinion, but the principle can be used for all kinds of stuff when trying to calculate quickly.
  For instance 9x =10x-x is usually faster than 9x (at least for my brain).
  I once talked to an old guy who called it "little math", because it fits in your head instead of having to use paper and pencil at the desk. It must have been taught differently before I was born. I work with numbers, and I've often encountered these old geezers who can eyeball a number close enough to make a decision before I can boot my pc and put everything through Excel.

2^{-2 * (2}-1)^{-1 = 2}-2 * 2^{(-1 * -1) = 2}(-2 + -1 * -1) = 2^{(-2 + 1) = 2}-1 = 1/2 = 0.5

Man, I thought this is an ADHD meme when trying to sleep and your brain starts to do random shit.

,25/,5 x 100/100 = 25/50 = 1/2

Why was that hard?

100/100 = 1, because any number divided by itself is 1.

And any number multiplied by 1 is still that number.

TBH, I moved the decimal over 2 places on the numerator and denominator and simplified 25/50 to 1/2 because It is easier to do in my head. Some of the other paths are too complicated when I am going to sleep.

(1/4) ÷ 1/2 = 2/4 = 1/2 = 0.5

N / (1/2) = 2N/1 = 2N

Please use parenthesis to avoid ambiguity. N/1/2 could also mean N/2...
(Also I think you made a typo "2N/2" should be "2N/1")
- thank you very much

That's 2x2=4 level of stupid.

Shut up and take my upvote!

You have one apple. You divide it into quarters, so that you have 0.25 of an apple. Now divide it in half. So yes well technically you do have one half of 0.25 (and 0.5 is the answer that a calculator will return) what you actually have is 1/8 of an apple (0.125).

This is what pisses me off about Matt half the time. You end up with something that in the abstract makes sense because it's just numbers, but then if you try to make it make sense in real life it's stupid.

Your confusion comes from the fact that dividing a quarter in half is 0.25/2. That's not what's shown in the comic. Dividing by 0.5 is the same as multiplying by 2. It's just a quirk of syntax, the way we write math. If you spelled it out using English the comic would say "multiplying a quarter by two equals a half". The confusion just stems from someone's unfamiliarity with mathematical notation.
- exactly. failure of english, not math. Math allows fairly accurate descriptions of the universe, humans (and especially languages) evolved to adequately percieve the narrow band of qualia that have been relevant to survival
Fractions are just funky when dividing. Dividing by 0.5 is the same multiplying by 2.
Your analogy is really close, but backwards is all. If you have a quarter of an apple. In order to get half a whole apple, you need another quarter. Two quarters make a half, so dividing a quarter by 0.5 gives you 2 quarters. Dividing a quarter by 2 gives you 0.5 of the original quarter which is your 1/8th