A number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.
On a number line, position shows size: anything to the left is smaller. The space between two points is their distance, here 4 − (−3) = 7.
Formal definition and notation
A number line is a line on which a point called the origin is labeled 0 and a fixed unit length is chosen. Each real numbera matches exactly one point: the point |a| units from 0, to the right if a > 0 and to the left if a < 0. For numbersa and b, a < b exactly when a lies to the left of b, and the distance between them is |a − b|.
Symbol
Read it as
Meaning
a < b
a is less than b
a is to the left of b on the number line
|a − b|
the absolute value of a minus b
the distance between the points a and b
● and ○
filled dot, open dot
on a graph of an inequality: endpoint included or not included
Key facts
The spaces between tick marks must be equal. Each space can stand for 1, 2, 10, 0.1 or any other step, but the step cannot change along the line.
Every integer, fraction, decimal and irrational number has a place: 34 is three of four equal parts between 0 and 1, and √2 ≈ 1.41 is a bit less than halfway from 1 to 2.
A number line is one dimension: one number names a point, such as −3. A coordinate plane uses two number lines, so it takes two numbers, such as (−3, 2).
A line plot is a data display that stacks an X or dot above a number line for each data value. The number line is just the scale underneath it.
Quick check: 3 questions
Question 1 of 3
Which number is farthest to the left on a number line?
Show answer and explanation
Answer: C. The farthest left is the smallest number, −9. For negative numbers, a larger absolute value means farther left. 0 and −1 are close to the middle, and −4 is to the right of −9.
Question 2 of 3
What is the distance between −3 and 8 on a number line?
Show answer and explanation
Answer: B.8 − (−3) = 11. 5 subtracts 3 from 8 and forgets the negative sign, −11 is not possible because distance is never negative, and 24 multiplies.
Question 3 of 3
A number line has tick marks at 0 and 1 with the space between split into 5 equal parts. What number is at the third mark after 0?
Show answer and explanation
Answer: D. Each part is 15, so three parts is 35. 13 and 34 use the wrong number of parts, and 3 counts parts as whole units.
Number Line: frequently asked questions
How do you put fractions on a number line?
Split each unit, the space from one whole number to the next, into as many equal parts as the denominator, then count parts equal to the numerator. For 34, divide the space from 0 to 1 into 4 equal parts and count 3 of them. For 213, go to 2, split the next unit into thirds and count 1.
Why do numbers get smaller to the left?
That is how the line is set up: 0 in the middle, and each step right adds 1. Going left subtracts 1 each step, so after 0 come −1, −2, −3. That is why −10 is less than −2: it is ten steps below 0, not two. Owing 10 dollars is worse than owing 2 dollars.
How do you add and subtract on a number line?
Start at the first number. To add a positive number, move right that many units; to subtract one, move left. To add a negative number, move left. For −4 + 6, start at −4 and move 6 right to land on 2. For 3 − 5, start at 3 and move 5 left to land on −2.
LineA line is a perfectly straight path of points that goes on forever in both directions. It has no thickness and no endpoints, so it has no length you can measure. Through any two points there is exactly one line, which is named by those two points, as in line AB, or by one lowercase letter.Geometry · Grade 4, Grade 8, GeometryOriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8Irrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1OppositeThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Number System · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Line PlotA line plot shows data values along a number line, often with marks above each value. Each X or dot stands for one data value, and repeated values stack in a column, so the tallest stack is the mode. Line plots make it easy to see clusters, gaps, outliers and the spread of a small data set.Statistics · Grade 3, Grade 4, Grade 5DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4, Grade 5Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1Interval NotationInterval notation is a short way to write a set of numbers between two endpoints, using brackets and parentheses. A bracket, [ or ], means the endpoint is included; a parenthesis, ( or ), means it is not. So −2 ≤ x < 3 is written [−2, 3), and x > 4 is written (4, ∞).Symbols and Notation · Algebra 1, Algebra 2, Precalculus