real numbers
Real numbers are a mathematical concept that encompass all the numbers that can be expressed on the number line
Real numbers are a mathematical concept that encompass all the numbers that can be expressed on the number line. Real numbers include both rational and irrational numbers. The set of real numbers is denoted by the symbol R.
Rational numbers are those that can be expressed as a fraction or ratio of two integers. This includes whole numbers, integers, and fractions. For example, 3, -5, 2/3, and -7/9 are rational numbers. Rational numbers can be represented as terminating decimals or repeating decimals. For instance, the fraction 2/3 can be written as the decimal 0.666… where the digit 6 repeats infinitely.
Irrational numbers, on the other hand, cannot be expressed as a fraction or ratio of two integers. They exist outside the realm of rational numbers. Examples of irrational numbers include square roots of non-perfect squares such as √2, π (pi), or e (Euler’s number). Irrational numbers cannot be represented as terminating or repeating decimals. For example, the square root of 2 is approximately 1.41421356… but the digits after the decimal point continue without repeating in a pattern.
Together, rational and irrational numbers form the set of real numbers. The real numbers have many important properties, such as closure under addition, subtraction, multiplication, and division. They can also be compared using the concepts of greater than, less than, and equal to. The set of real numbers plays a fundamental role in various branches of mathematics, including algebra, calculus, and geometry.
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Understanding Irrational Numbers | Definitions, Examples, and Importance in Mathematics