questions including [x]

Question

does anyone know what   means?

eg: If   denotes the greatest integer less than or equal to x, is   = 0

I'm having trouble understanding what  If   denotes the greatest integer less than or equal to x  means

rjacobsMGMAT · Answer

Well it looks to me like denotes the greatest integer less than or equal to x.

("denotes" = "is") It's a function that the GMAT made up.

EvaJager · Answer

It's not a function made up by GMAT

) Carl Friedrich Gauss introduced the square bracket notation for the floor function, which indeed, associates to every x the greatest integer less than or equal to x. Much later, in computer science, Iverson introduced different notations for the floor \lfloor{x}\rfloor (and ceiling \lceil{x}\rceil ) function. In mathematics both notations are used.

EvaJager · Answer

Every real number can be placed between two consecutive integers, such that  k\leq{x}<k+1 . 
For every given  x , the integer  k with the above property is unique. By definition, then    = k .
You can always write   \leq{x}< +1. 
If  x=5,    =5 . If  x=5.67 ,   =5. 
  =0 ,   =0 , but   =-1.

If  x  is an integer, than     is simply  x. 
For non-integer  x , try to visualize the number line. The closest integer to  x  from the left is   . 
We used to call it  the integer part function . For positive numbers, just ignore the decimal part and leave the integer part of the number.
For negative numbers, don't forget to look at the left of your number.    = -3  and not  -2.

So, if    = 0 , then certainly  0\leq{x}<1 , meaning  x  can be any number between 0 and 1, 0 inclusive, 1 exclusive.

rjacobsMGMAT · Answer

HA that's great, I totally didn't know that! I usually teach my students that this is the equivalent of the RoundDown function on Excel, but I think I like this history lesson better

EvaJager · Answer

I know this is not the place, but I see I can't send you a PM (personal message) through the site. Thank you for the Kudos, but I think I didn't do much to earn it in this case:o) BTW, I thoroughly enjoyed your post "Winning ugly on the GMAT", Kudos for it! I cannot deny Agassi's achievements, but my all times favorites is still Sampras. Although he never won Roland Garros, he had a superb game at the net, which today's player lack so much. Definitely, Agassi made the maximum of what he had at hand. Great message and amazing idea to connect tennis with GMAT! For those who didn't read the article yet: https://www.manhattangmat.com/blog/index ... -the-gmat/

Bunuel · Answer

Questions testing this function to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
is-x-an-odd-integer-1-x-2-2x-99856.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html
for-all-z-denotes-the-least-integer-greater-than-or-110561.html

KarishmaB · Answer

When working on Floor or Ceiling functions, a useful technique is to think of the number line.

Say x lies somewhere on the number line. The floor function, often denoted by  , is the integer closest to x on the left hand side i.e. it is the largest number which is smaller than x (or equal to x if x is an integer). It is easy to comprehend in case of positive numbers. e.g. 
  = 3, 
  = 5 etc
You should be a little more careful while dealing with negative numbers
  = -3
This is why the number line perspective helps.

bumpbot · Answer

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