Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I need help with the definition of "within 1":

  • If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$.

  • If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$.

  • If $x = 8$ and $y = 8$, is $x$ still "within 1" of $y$?

It's my understanding that this would still be true, but I'm being asked for something to back up my assumption, so I guess I'm looking for a second opinion.

share|improve this question
Usually, this terminology is used in a context where you have units. "Within 3 meters of the target", "within 1 ULP of the exact result", "within 0.01 K of absolute zero". –  user2357112 Jun 9 at 6:03

5 Answers 5

up vote 4 down vote accepted

It means that the value lies within the limits of +/− 1.

If you were to say 'within 1' of 20, that means that 19, 20, and 21 are all valid numbers because they're 'within 1' of 20.

The most common term for this is 'plus or minus 1' or whatever range you're looking in. Symbols used to denote this particular range are +/− and ±. This is used a good bit in statistics and the sciences.

share|improve this answer
I'd also add that the use of $\pm$ to mean "within" is less common in mathematics, where $\pm 1$ is generally used to mean "+1 or -1" and no possibilities in between (e.g. in the quadratic formula). –  Erick Wong Jun 8 at 19:36

In this case, it probably means that the absolute value of the difference between the two numbers does not exceed $1$. Hence $8$ is within $1$ of $8$. Note that this expression does not occur frequently in mathematical literature.

share|improve this answer

"Within $x$" refers to $\pm x$. Hence, given a number $y$, the numbers within $x$ of $y$ are elements of the set $$Z=\{z\mid y-x\leq z\leq y+ x\}$$ Quite obviously, since $y-x\leq y\leq y+x$, $y\in Z$.

share|improve this answer

In the more general case, I would say that "$x$ is within (a distance) $d$ of $y$" means that $$|x - y| \le d.$$ (Depending on the context, I would imagine the inequality could be strict.)

share|improve this answer

I have come across a similar query around the definition of "within". The dictionary definition (from www.oed.com) is "That which is within or inside".
Given the "or" in the definition the "within 1 of" includes the value 1.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.