# What is the definition of 'within one' in mathematics

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.

• 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". Jun 9, 2014 at 6:03

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.

• 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). Jun 8, 2014 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.

"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 $y-x\leq y\leq y+x$, $y\in Z$.

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.)

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.