The greatest integer function, explained from scratch

What the function does to a number

Read $\lfloor x\rfloor$ as an instruction: stand at $x$ on the number line and walk left until you reach an integer. If you are standing on one already, stay where you are.

• $\lfloor 3.7\rfloor =3$. The integers at or below 3.7 are $\dots ,1,2,3$, and the greatest of them is 3.
• $\lfloor 4\rfloor =4$. The definition says less than or equal to, so 4 is its own answer. The function leaves every integer where it is.
• $\lfloor -3.7\rfloor =-4$. Walking left from $-3.7$, the first integer you meet is $-4$. The integer $-3$ lies to the right of $-3.7$, so it is too big to count.

An integer plus a leftover

A second way to see it settles negative numbers for good. Every real number can be split, in exactly one way, into an integer and a leftover that is at least 0 and less than 1:

The integer $n$ is $\lfloor x\rfloor$. For the first two numbers the split is plain: $3.7=3+0.7$ and $4=4+0$. The negative one is where it earns its keep: $-3.7=-4+0.3$, so $\lfloor -3.7\rfloor =-4$. Splitting $-3.7$ into $-3$ and a leftover of $-0.7$ does not count, because the leftover has to be at least 0.

Written as inequalities, the same fact reads: $\lfloor x\rfloor =n$ exactly when $n\le x<n+1$. Both forms are used below.

A word on notation. The ministry's notation guide for secondary mathematics writes the greatest integer of $a$ with square brackets, $[a]$, so you may meet that form in class. This site writes $\lfloor a\rfloor$, as its worksheets do. The two mean exactly the same function.

Why the graph is a staircase

1. Nothing changes between two integers. Every $x$ in $[3,4[$ splits as $3+d$ with $0\le d<1$, so $\lfloor x\rfloor =3$ on that whole interval. Over it, the graph is a flat segment at height 3.
2. At each integer the output jumps by 1. $\lfloor 3.99\rfloor =3$, but $\lfloor 4\rfloor =4$.
3. The jump decides the dots. The point $(4,4)$ is on the graph, so the step at height 4 begins with a closed dot. The point $(4,3)$ is not, because $\lfloor 4\rfloor$ is 4, so the step at height 3 ends with an open dot: it runs right up to 4 without reaching it.
4. So every step is half-open. For $y=\lfloor x\rfloor$ the steps are $[n,n+1[$: one unit long, one unit above the step before, closed on the left and open on the right.

Worked example 1: every number that goes to −4

Find every $x$ with $\lfloor x\rfloor =-4$.

1. By the definition, $\lfloor x\rfloor =-4$ means $-4\le x<-4+1$.
2. So $-4\le x<-3$, that is, $x\in [-4,-3[$.
3. Check both ends. $\lfloor -4\rfloor =-4$, so $-4$ belongs. $\lfloor -3\rfloor =-3$, so $-3$ does not.
4. Check a point inside: $-3.7$ is in the interval, and $\lfloor -3.7\rfloor =-4$, as found above.

The answer is the bottom step of the figure, closed at its left end and open at its right.

Why it models anything charged per started block

On every step a whole interval of inputs shares one output, and that is exactly how a price paid by the block behaves: flat while you stay inside one block, then a jump the moment the next one begins. Take a parking lot that charges $3 for each started 20 minutes.

Worked example 2: the parking rule

Write the cost $C(t)$, in dollars, of a stay of $t$ minutes, $t>0$.

1. Count blocks by hand first. A 5-minute stay starts one block. So does a 20-minute stay: it fills the first block exactly and starts no second one. A 21-minute stay starts two blocks, a 40-minute stay still two, a 41-minute stay three.
2. Turn the count into inequalities. A stay that starts $n$ blocks is longer than $n-1$ blocks and no longer than $n$ blocks: $$n-1<\frac{t}{20}\le n$$
3. Multiply every part by $-1$. Both inequality signs reverse: $$-n\le -\frac{t}{20}<-n+1$$
4. Recognize the definition. That is $m\le u<m+1$ with $u=-\frac{t}{20}$ and $m=-n$, so $$\lfloor -\frac{t}{20}\rfloor =-n,n=-\lfloor -\frac{t}{20}\rfloor$$
5. Price the blocks. At $3 a block: $$C(t)=-3\lfloor -\frac{t}{20}\rfloor$$
6. Test at a step end and just past it. $C(40)=-3\lfloor -2\rfloor =6$, and $C(40.5)=-3\lfloor -2.025\rfloor =-3(-3)=9$. Forty minutes is two blocks, $6; half a minute more starts a third, $9.

In the rule $f(x)=a\lfloor b(x-h)\rfloor +k$, this is $a=-3$, $b=-\frac{1}{20}$, $h=0$ and $k=0$. The negative $b$ is what moves the closed dot to the right end of every step: the $6 step is $]20,40]$, because a 40-minute stay is still two blocks.

The plain function counts the other kind of block: $\lfloor \frac{t}{20}\rfloor$ is the number of completed 20-minute blocks, two in a 47-minute stay. Whole years of age and full boxes are counted that way; anything charged "per block or part of a block" needs the two minus signs.

Check your understanding

No calculator needed.

1. Evaluate $\lfloor 9.2\rfloor$, $\lfloor -9.2\rfloor$ and $\lfloor -9\rfloor$.
2. Find $\lfloor \pi \rfloor$ and $\lfloor -\sqrt{2}\rfloor$.
3. Which of these points lie on the graph of $y=\lfloor x\rfloor$: $(2,1)$, $(2,2)$, $(1.99,1)$ and $(-0.1,0)$?
4. How many multiples of 7 are there from 1 to 100? Show that the count is $\lfloor \frac{100}{7}\rfloor$ and explain why.
5. True or false, with a reason: if $\lfloor x\rfloor =5$, then $\lfloor x+0.5\rfloor =5$.
6. Each situation counts blocks. Say whether it counts completed blocks or started blocks, and write the count with the greatest integer function. (a) A person's age in whole years, $t$ years after birth. (b) The number of 30-second blocks billed for a phone call of $s$ seconds, when every started block is billed. (c) The number of complete dozens in $e$ eggs.
7. A laundromat dryer costs $2 for every started 15 minutes. A student models a cycle of $t$ minutes with $D(t)=2\lfloor \frac{t}{15}\rfloor +2$, tests it at $t=10$ and at $t=25$, and both tests match the price list. Find a cycle length where the model is wrong, explain why the tests missed it, and give a correct rule.

Practise it

The free Secondary 4 step functions worksheet takes the staircase further: the parameters $a$, $b$, $h$ and $k$, graphing with the right dots, and pricing problems, with the methods explained on the page. Solving an equation such as $\lfloor x\rfloor =-4$ is practised in the Secondary 4 equations and inequalities worksheet.

Practise the greatest integer function

• Secondary 4 Step Functions worksheet
• Secondary 4 Equations and Inequalities worksheet