Real numbers: Definition and basics at a glance

Real numbers are an important part of mathematics. They are referred to as ℝ and include all the numbers we know. These numbers are important because they form a complete number line.

The definition of real numbers includes both rational and irrational numbers. This helps us to make precise calculations in many areas. For example, when we count the price of a chocolate bar, €1.95.

Real numbers have many subsets. These include rational, integer and natural numbers. But also an infinite number of irrational numbers such as π. This diversity enables precise mathematical work in our world.

The basics of real numbers

Real numbers are the foundation of mathematics. They are used in many areas. They include rational and irrational numbers and are important in many professions.

Definition and mathematical meaning

Real numbers are complete and ordered. They are based on 15 axioms. Of these, 9 are arithmetic axioms and 4 are ordering axioms.

These axioms define the properties of real numbers. They enable precise mathematical operations.

Properties of real numbers

The most important properties include

• Density: There is always another real number between two real numbers
• Completeness: Every finite set of real numbers has a supremum and infimum
• Order structure: Real numbers can be ordered

These properties enable precise calculations in various areas.

Number line and representation

The number line is a visual representation of real numbers. It shows their arrangement and enables the representation of intervals. Both rational and irrational numbers can be represented on the number line.

Real numbers are used in many professions. From architects to financial advisors. They enable complex calculations such as trajectories or interest calculations. They are essential for precise mathematical models in science and technology.

Rational and irrational numbers as components

Real numbers consist of rational and irrational numbers. These two types are important for many mathematical calculations. They are used in many areas.

Characteristics of rational numbers

Rational numbers can be represented as a fraction of two whole numbers. They include whole, natural and decimal numbers. There is always another number between two rational numbers.

Special features of irrational numbers

Irrational numbers cannot be represented as a fraction of two whole numbers. Their decimal expansion is infinite and non-recurring. Examples are π and e. There are an infinite number of irrational numbers between two rational numbers.

Distinguishing features and examples

The main thing is whether they can be represented as a fraction. Rational numbers such as 1/2 or 3.14 can be represented as a fraction. Irrational numbers such as √2 or π cannot be represented as a fraction. Their decimal expansion is infinite and non-recurring.

Mathematical operations with real numbers

Real numbers enable many mathematical operations. You can add, subtract, multiply and divide them. There are also more complex operations such as exponentiation and root extraction.

These calculations are important in architecture, engineering and finance.

Conclusion

Real numbers are very important in math and science. They include rational and irrational numbers. Irrational numbers cannot be represented as a fraction.

The set ℝ arises from ℚ. It is a model for a continuous linear continuum.

Real numbers have many applications. In financial mathematics, they help to calculate interest and investments precisely. Engineers use them to determine forces and loads.

In computer science, the IEEE-754 standard shows how to represent real numbers.

Real numbers are important in analysis, topology and geometry. They help to model continuous processes. They are the basis for many advanced mathematical ideas.

The completeness of real numbers shows their central importance. They have a supremum for every non-empty, upper bounded subset.