I know people here might find it silly but I want to clear my understanding about remainder.

I had a fight with my brother over this question. I suggested that remainder is something that is left when two numbers can't further be divided.

This is what I think remainder should be:- $$\dfrac{318}{100}=3+\dfrac{18}{100}$$ $$\dfrac{318}{100}=3+\dfrac{9}{50}$$So according to me 9 should be the remainder of $\dfrac{318}{100}$ as 9 cannot further be divided by $50$.

This is what my brother does enter image description here

So according to him remainder should be $18$. But this is not the remainder actually this something we call mod of two number (which is usually used by computers to calculate remainder of two numbers ). My brother says that 9 is not the remainder of $\dfrac{318}{100}$ but the remainder of $\dfrac{159}{50}$. Aren't they the same thing.

Can anyone tell me, who is correct.

  • $\begingroup$ "The remainder is what is left when two numbers can't be divided further" : this ambiguous statement is your downfall. Also, the "mod" of two numbers is the remainder when one is divided by the other. Your brother is correct. $\endgroup$ – астон вілла олоф мэллбэрг Sep 26 '18 at 4:39
  • $\begingroup$ Why the downvote? Is it inappropriate to ask for clarification on the definition of the remainder of one integer when divided by another? $\endgroup$ – eloiprime Sep 26 '18 at 4:45

The remainder of $n$ divided by $d$ is the unique integer $r$ satisfying $0\le r<d$ and $n=qd+r$ for some integer $q$. When $n= 318$ and $d=100$, $r=18$ satisfies the criterion (with $q=3$).

  • 2
    $\begingroup$ And to rearrange your equation in a form that may be relevant to the OP, it's specifically $$\frac{n}{d} = q + \frac{r}{d}$$ where both denominators are the same. $\endgroup$ – Hurkyl Sep 26 '18 at 4:41


2 b (2) : the final undivided part after division that is less or of lower degree than the divisor


As an employee of a storefront tutoring/enrichment centre, I can confirm that children are taught to notate remainders as integers. It’s just terminology. When I say “$100$ divides $318$ to $3$ with remainder $18$,” you know exactly what I mean: $18$ is leftover. This comes from a very concrete and rigid approach to division, like evenly allocating students (which can’t be cut in two) with field trip tour guides (or what have you).

Then, when you get into higher maths, you have to execute the division process fully—like you did. This, by its very notion, leaves no remainder. Instead of an integral remainder, we’re left with a decimal or mixed number.

So when you have a non-integer as your quotient, that $9/50$ is not the remainder but rather the fractional part of the quotient. (We often notate the fractional part of a number $x$ as $\{x\}$.)

  • $\begingroup$ I take objection to your description of division in higher mathematics. In higher mathematics, we generalize the notion of the division algorithm found in the integers to other structures (called Euclidean domains). Once we start "executing the division process fully", as you say, we are effectively working in a field. In fields, the notion of division is no longer useful. Indeed, remainders are always $0$. $\endgroup$ – eloiprime Sep 26 '18 at 5:20
  • $\begingroup$ @eloiPrime Of course you object—my answer is competing with yours. I did not specify how high the mathematics is though, just that it’s higher than elementary school. $\endgroup$ – gen-z ready to perish Sep 26 '18 at 5:36

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