Number basics
The Number Line
A number line is a straight line with numbers placed at equal intervals: 0 in the middle, positive numbers to the right and negative numbers to the left. Every number has exactly one spot on it. This guide covers how to read a number line’s scale, plot integers, fractions and decimals, compare and order numbers, add and subtract by moving along it, and measure distance between two points.
What a number line shows
A number line turns numbers into positions. It has four parts:
- The origin, 0, the reference point.
- Tick marks at equal spacing. Equal spacing is the whole point: the gap from 2 to 3 must look the same as the gap from −5 to −4.
- Labels on some or all ticks.
- Arrows at each end, showing the line keeps going forever in both directions.
Right means bigger, left means smaller. That one rule explains ordering, adding, subtracting and why −10 is less than −1. The number line is also where negative number rules stop being rules to memorize and become directions you can see.
Not every number line starts at 0 or shows negatives. A line for a thermometer reading might run from 60 to 80; a line for fractions might run only from 0 to 1. What stays fixed is that each step of the same size covers the same distance.
How to read a number line
When only some tick marks are labeled, work out what each step is worth before you read anything.
Example: labels 20 and 50, six gaps between them
A point sits on the third tick after 20.
Example: labels 0 and 1, four gaps between them
The middle tick is 2/4, which simplifies to 1/2. The same position can carry several names — the guide to equivalent fractions explains why.
Plotting integers, fractions and decimals
Integers
Integers sit exactly on tick marks when the scale goes up by 1. To plot −4, start at 0 and count four steps left. To plot 6, count six steps right.
Fractions
Split each unit into as many equal parts as the denominator, then count as many parts as the numerator. For 3/4, split 0 to 1 into 4 parts and count 3. For an improper fraction such as 7/3, convert to a mixed number first: 7/3 = 2 1/3, so go to 2, split the next unit into thirds, and count one third further.
Decimals
Treat decimals as fractions of 10, 100 and so on. For 2.6, split 2 to 3 into ten parts and count six. If the fraction is awkward to place, the fraction to decimal calculator converts it so you can locate it by size.
Negative fractions and decimals
Mirror the positive version across 0. −1 1/2 is the same distance left of 0 as 1 1/2 is to the right, which puts it halfway between −1 and −2. A common slip is placing −1 1/2 between −1 and 0; that would be −1/2.
Numbers between the ticks
The tick marks are only signposts. Every point on the line is a number, including all the rational numbers in between and irrational ones such as √2 ≈ 1.414, which sits a little less than halfway from 1 to 2. The whole numbers are just the points 0, 1, 2, 3 and onward, and the integers add their mirror images to the left. However closely you zoom in, there are always more numbers between any two points.
Example: plot −2, −0.75, 1/2 and 1.8
Read left to right, the order is −2, −0.75, 1/2, 1.8.
Comparing and ordering numbers
The number further right is greater. That holds for every kind of number, and it settles the comparisons that feel backward with negatives.
Example: order −3, 1/2, −0.5, 2 and −3/2
Convert to decimals so they can share one line, then read left to right.
Inequalities on a number line
A number line can show a whole range of numbers at once. Shade the part of the line that satisfies the inequality and mark the endpoint with a circle.
- Open circle for
<or>: the endpoint is not included.x > 2is an open circle at 2 with the line shaded to the right. - Closed (filled) circle for
≤or≥: the endpoint is included.x ≤ −1is a filled circle at −1 with the line shaded to the left.
Adding and subtracting on a number line
Start at the first number. Adding a positive number moves you right; adding a negative moves you left. Subtracting reverses the direction: subtracting a positive moves left, and subtracting a negative moves right.
| Problem | Start | Move | Land on |
|---|---|---|---|
| −3 + 5 | −3 | 5 right | 2 |
| 4 − 7 | 4 | 7 left | −3 |
| −1 + (−2) | −1 | 2 left | −3 |
| −2 − (−6) | −2 | 6 right | 4 |
| 0.8 + (−1.3) | 0.8 | 1.3 left | −0.5 |
Why does subtracting a negative move right? Subtraction asks “how far is it from the second number to the first?” From −6 to −2 is 4 steps to the right, so −2 − (−6) = 4. The negative number rules guide works through every sign combination this way; the table above is enough to see the pattern.
Distance between two numbers
The distance between two points is how many units apart they are, and it is never negative. Subtract and take the absolute value, which removes the sign.
The last line is the definition of absolute value: |x| is the distance from x to 0. That is why |−5| and |5| are both 5 — the two points sit the same distance from 0 on opposite sides.
The midpoint
The point exactly halfway between a and b is their average, (a + b) ÷ 2. Halfway between −4 and 3 is (−4 + 3) ÷ 2 = −0.5. Check it: −0.5 is 3.5 units from each end.
Multiplying and dividing with jumps
Multiplication is repeated jumps of equal size. 3 × 4 is three jumps of 4 from 0, landing on 12. 3 × (−2) is three jumps of 2 to the left, landing on −6. A multiplication chart lists the landing points for every pair up to 12.
Division asks how many jumps of one size fit into a distance. 12 ÷ 3 asks how many jumps of 3 get from 0 to 12: four. The picture makes fraction division intuitive: 2 ÷ 1/2 asks how many half-steps fit between 0 and 2, and the answer is 4. That is the idea behind the flip-and-multiply method in how to divide fractions.
Rounding with a number line
Rounding means finding which of two neighbors a number sits closer to. Mark the two candidates and see which one is nearer.
The halfway case is where conventions differ; the usual school rule rounds it up. The rounding calculator shows each rule side by side, and rounding to decimal places covers the decimal cases.
Common mistakes
- Counting ticks instead of gaps. Divide the labeled distance by the number of spaces between labels.
- Starting the count at 1. Moving 3 steps from −2 lands on 1, not 0. Count jumps, not the point you start on.
- Mirroring negatives wrongly. −1.5 is between −1 and −2, further from 0 than −1 is.
- Uneven spacing. On a hand-drawn line, cramped ticks make 7 look closer to 5 than to 9. Space every step equally.
- Mixing up open and closed circles. Open means the endpoint is excluded (
<,>); filled means it is included (≤,≥). - Calling a distance negative. Distances are always 0 or positive. Use absolute value.
Frequently asked questions
What is a number line?
A number line is a straight line on which every number has a position, with equal spacing between consecutive values. Zero sits at the reference point, positive numbers run to the right and negative numbers to the left, and arrows at the ends show it continues forever. It is used to compare, order, add and subtract numbers visually.
How do you put fractions on a number line?
Divide each whole unit into as many equal parts as the denominator, then count as many parts as the numerator from the left end of the unit. For 3/4, split 0 to 1 into 4 equal parts and count 3. For 7/3, rewrite it as 2 1/3, go to 2, and count one third further.
Where is −1/2 on a number line?
−1/2 is halfway between −1 and 0, to the left of 0. It is the mirror image of 1/2, the same distance from zero on the opposite side. As a decimal it is −0.5, so it is greater than −1 but less than 0.
Is −5 greater than −2?
No. −5 is less than −2. On a number line −5 sits further to the left, and numbers decrease as you move left. Written as an inequality, −5 < −2. A temperature of −5°F is colder than −2°F, which is a useful way to check.
How do you find the distance between two numbers on a number line?
Subtract one from the other and take the absolute value: distance = |a − b|. The distance between −4 and 3 is |−4 − 3| = 7. You can also count the unit steps from one point to the other. Distance is never negative, and the order of the two numbers does not matter.
What does an open circle on a number line mean?
An open circle means the endpoint is not included. It is used with strict inequalities, < and >. The graph of x > 2 has an open circle at 2 and shading to the right, because 2 itself does not satisfy the inequality. A filled circle means the endpoint is included, for ≤ and ≥.
How do you add negative numbers on a number line?
Start at the first number and move left by the size of the negative number. For −1 + (−2), start at −1 and move 2 steps left to land on −3. For 3 + (−5), start at 3 and move 5 steps left to land on −2. Adding a negative always moves you left.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/number-line/