On the number line and in everyday measurements, real numbers are the coordinates, counts, and values that describe the world around us—used in everything from simple budgets to geometry and physics. Thinking in terms of location on a line makes it easy to spot patterns and relationships among them.
There are 20 Examples of Real Numbers, ranging from -1 to ∛2. For each entry the data are organized with column headings Decimal form,Classification,Where found, so you can compare exact values, type (integer, rational, irrational, etc.), and common contexts; see the list you’ll find below.
How can I tell whether a number in the list is rational or irrational?
Look at its decimal form: a terminating or repeating decimal is rational; a non-repeating, non-terminating decimal is irrational. The Classification column will state this directly, and simple tests (like whether a root simplifies to a fraction) help spot irrationals such as ∛2.
Why include both -1 and ∛2 among the examples?
They illustrate the range of real numbers: -1 shows negatives and integers, while ∛2 is an irrational root—together they demonstrate that real numbers include negatives, positives, integers, rationals, and irrationals, which the Decimal form,Classification,Where found columns will make clear.
Examples of Real Numbers
| Number | Decimal form | Classification | Where found |
|---|---|---|---|
| 0 | 0.00 | Integer | Counting, identity element |
| 1 | 1.00 | Integer | Counting, multiplicative identity |
| -1 | -1.00 | Integer | Algebra, opposites, finances |
| 1/2 | 0.50 | Rational | Fractions, probability, measurements |
| 1/3 | 0.33… | Rational | Repeating decimals, ratios, probability |
| 3/4 | 0.75 | Rational | Measurements, proportions, finance |
| 1/4 | 0.25 | Rational | Money, fractions, geometry |
| 1/7 | 0.14… | Rational | Repeating decimals, calendars, fractions |
| 22/7 | 3.14… | Rational | Pi approximation, elementary math |
| π | 3.14… | Transcendental | Geometry, circles, trigonometry |
| e | 2.72… | Transcendental | Calculus, growth, compound interest |
| √2 | 1.41… | Algebraic | Right triangles, Pythagorean theorem |
| √3 | 1.73… | Algebraic | Equilateral triangles, geometry |
| φ (golden ratio) | 1.62… | Algebraic | Art, Fibonacci, proportions |
| ∛2 | 1.26… | Algebraic | Geometry, cube roots, classic problems |
| Liouville constant | 0.11… | Transcendental | Constructive examples, transcendence proofs |
| Apéry’s constant ζ(3) | 1.20… | Irrational | Number theory, zeta function |
| 1,000 | 1,000.00 | Integer | Counting, rounding, magnitudes |
| -273.15 | -273.15 | Rational | Temperatures, Celsius absolute zero |
| 0.25 | 0.25 | Rational | Percentages, finance, probabilities |
Images and Descriptions
0
Zero is the additive identity in arithmetic and algebra. It’s a real integer used across math and everyday counts, marking the absence of quantity and enabling place-value and algebraic structure.
1
One is the multiplicative identity and a fundamental integer. It appears in counting, units, and scaling; as a real number it serves as the simplest nonzero value in arithmetic and algebra.
-1
Minus one is an integer representing the additive inverse of one. It is a real number used in algebra, sign changes, and modeling debts or losses in practical contexts.
1/2
One half is a simple rational number equal to 0.5. It commonly represents splitting things in two, probability 50%, and appears in measurements and basic algebra as an exact real value.
1/3
One third is a rational number with the repeating decimal 0.33… It models equal partition into three parts, appears in ratios and probability, and is exactly representable as a fraction.
3/4
Three quarters equals 0.75 and is a rational number. It shows up in recipes, percentages, and proportional reasoning as a precise real value derived from division of integers.
1/4
One quarter is 0.25, a terminating rational decimal common in money, measurements, and geometric fractions. It’s a real number exactly equal to 1 divided by 4.
1/7
One seventh is a rational number with a repeating decimal 0.14… (period six). It’s often used in fractional schedules and as a classic example of repeating decimal behavior.
22/7
Twenty-two sevenths is a simple rational approximation to pi (3.14…). It’s historically used for rough circle calculations and shows how rationals can approximate irrational constants.
π
Pi is the transcendental constant relating a circle’s circumference to its diameter. Its nonterminating, nonrepeating decimal expansion makes it a real number central to geometry, physics, and engineering.
e
Euler’s number e is a transcendental constant arising from continuous growth and calculus. Its decimal expansion doesn’t repeat, and it appears in natural logarithms, compound interest, and differential equations.
√2
The square root of two is an irrational algebraic number solving x^2=2. It appears as the diagonal length of a unit square and was historically the first number proven irrational.
√3
Square root of three is an algebraic irrational from x^2=3. It appears in geometry of equilateral triangles and trigonometry and has a nonrepeating decimal expansion, making it a real but irrational number.
φ (golden ratio)
The golden ratio φ solves x^2−x−1=0 and is an algebraic irrational about 1.62… It shows up in Fibonacci limits, aesthetics, and certain geometric proportions in nature and design.
∛2
The cube root of two is an algebraic real solving x^3=2. Historically tied to the doubling-the-cube problem, it’s irrational and appears where cubic measures or scaling are needed.
Liouville constant
Liouville’s constant is a constructed real number with a carefully chosen decimal expansion that proves the existence of transcendental numbers. Its nonrepeating decimals make it transcendental by design.
Apéry’s constant ζ(3)
Apéry’s constant is the value ζ(3) of the Riemann zeta function; it’s known to be irrational. It appears in series, number theory, and some physics sums.
1,000
One thousand is a round integer used for counting and scaling. As a real number it’s exact, frequently used in statistics, finance, and everyday approximations for large counts.
-273.15
Negative 273.15 degrees Celsius is the defined value for absolute zero on the Celsius scale. It’s a terminating decimal rational number used in thermodynamics and temperature conversions.
0.25
Zero point two five is a terminating rational decimal equal to one quarter. It’s widely used in percentages (25%), finances, and simple probability statements as an exact real value.
