Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration or temperature. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus and in many other branches of mathematics, especially in the classical definitions of limits, continuity and derivatives.

The set of real numbers includes the rational numbers, such as the integer −5 and the fraction 4 / 3, as well as numbers that are not rational, known as irrational numbers. Examples of irrational numbers include √2 and π. Real numbers can be visualized as points on a line called the number line or real line.

The concept of real numbers was developed in the 19th-century mathematics to allow for rigorous reasoning in mathematics. Today, real numbers form the basis of real analysis, the study of real functions and real-valued sequences. They are essential for understanding many aspects of the world around us, from physics to engineering.

Characterizing properties

Real numbers are special because they follow certain rules that make them easy to work with in math. These rules let mathematicians and scientists use real numbers for calculations even though they didn’t fully understand what real numbers were made of until the 1800s.

Because of these rules, all real numbers behave the same way, which helps us solve problems and understand measurements like length or temperature.

Arithmetic

The real numbers are a system where we can perform basic math operations like addition and multiplication. These operations follow specific rules that make calculations consistent and predictable.

For example, adding two real numbers always gives another real number, and the order of addition doesn't matter (such as 3 + 4 being the same as 4 + 3). Similar rules apply to multiplication, and these operations work together in special ways, like how multiplying a sum can be broken into smaller parts. Real numbers also include special values like zero (which doesn't change a number when added) and one (which doesn't change a number when multiplied).

Topological completeness

Main article: Completeness of the real numbers

Real numbers are special because many sequences of numbers have limits. This means that the numbers in the sequence get closer and closer to a specific value.

For example, the sequence that adds more digits of the square root of 2 — like 1, 1.4, 1.41, 1.414, and so on — gets closer to the square root of 2, even though it never exactly reaches it. This shows that real numbers are "complete," meaning there are no gaps. This idea of completeness is important in math, especially in calculus and analysis.

Cardinality

The set of all real numbers is uncountable. This means that even though both the set of all natural numbers like 1, 2, 3, and the set of all real numbers are very large, there is no way to pair every real number with a unique natural number. The size, or cardinality, of all real numbers is called the cardinality of the continuum, and it is bigger than the size of the natural numbers, which is called aleph-zero.

Mathematicians have a question called the continuum hypothesis about whether there is a size between these two. The most common system of rules for mathematics, called Zermelo-Fraenkel set theory with the axiom of choice (ZFC), cannot prove or disprove this hypothesis. Some versions of these rules support the hypothesis, while others do not.

Other properties

See also: Real line

Real numbers have many important properties. They can be thought of as points on a line, where the distance between any two points can be measured. This makes them useful for describing distances and measurements in the real world.

Every positive real number has a square root, which is another real number. This helps us solve equations and understand patterns in numbers. Real numbers also follow special rules that make them easy to work with in mathematics.

History

Simple fractions were used by the Egyptians around 1000 BC. Around the same time, early Indian mathematicians began to understand that some numbers, like the square roots of 2, could not be easily written as fractions.

Later, Greek mathematicians also discovered that the square root of 2 was one of these special numbers that could not be written as a simple fraction. Over time, different cultures developed ways to work with these numbers, eventually leading to the modern idea of real numbers we use today.

Formal definitions

Main article: Construction of the real numbers

Real numbers are a special kind of number used in math. They help us describe things that can be measured exactly, like the length of a line or the temperature outside. These numbers can be very precise, and we can write them using endless decimals, like 3.14159... for pi.

All real numbers follow certain rules, and these rules make sure that real numbers behave in a consistent way. No matter how you build these numbers, the rules stay the same, which means we can think of all real numbers as being part of one big, orderly system.

Applications and connections

Real numbers are important in science. They help describe things like how far an object is, how heavy it is, or how fast it moves. Big ideas in physics, like how gravity works or how tiny particles behave, use real numbers to make their theories work.

Computers can’t handle all real numbers perfectly because they would need endless digits. Instead, they use approximations, which can sometimes cause small errors in calculations. Special tools can handle some real numbers exactly, but this can be slow and use a lot of memory.

Vocabulary and notation

The set of all real numbers is written as R or ℝ. These numbers can describe things like length or temperature.

We also have special groups of real numbers, like positive real numbers (ℝ⁺) and negative real numbers (ℝ⁻). When we write ℝⁿ, we mean sets of n real numbers together, like points in space.

Generalizations and extensions

The real numbers can be expanded in many ways. One way is through complex numbers, which solve all polynomial equations but cannot be ordered like real numbers.

Other expansions include the affinely extended real number system, which adds two special values, +∞ and −∞, and the real projective line, which adds a single value ∞ and allows division by zero. These systems are not fields like the real numbers but have interesting properties.

There are also larger systems like the long real line, which is like the real numbers but extended to include many more points. Other extensions, such as hyperreal numbers and surreal numbers, include very small and very large numbers that are not possible in the real numbers.

[Self-adjoint operators] on a Hilbert space also generalize real numbers in many ways, maintaining order and completeness while corresponding to real values in certain situations.

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