Why do we need rational and irrational numbers?

Question

Autumn Arnold · Accepted Answer

See explanation.

The purpose of all real number subsets is to increase the range of mathematical operations that can be applied to them. First set was natural numbers (#NN#) . Only addition and multiplication could be performed on this set. To make substraction possible people had to invent negative numbers and expand natural numbers to integer numbers (#ZZ#) While division operations were not possible in this set, multiplication, addition, and subtraction were. To extend the range to all 4 basic operations (addition, substraction, multiplication and division) this set had to be extended to set of rational numbers (#QQ#) However, not all operations could be completed even with this set of numbers. If we try to calculate the hypothenuse of an isosceles right triangle, whose catheti have length of #1# we get a number #sqrt(2)# which is an example of irrational number. If we add rational and irrational numbers we get the whole set of real numbers (#RR#)

Gianna Caruso · Answer

undefined Rational numbers are needed to represent quantities that can be expressed as a ratio of two integers, such as fractions or terminating decimals. Irrational numbers are needed to represent quantities that cannot be expressed as a simple fraction, such as the square root of non-perfect squares or transcendental numbers like π and e. Together, rational and irrational numbers provide a complete and continuous number line, allowing us to represent and manipulate a wide range of quantities in mathematics and real-world applications.

Autumn Arnold · Answer

undefined We need rational numbers because they represent quantities that can be expressed as fractions of integers. They are essential for practical applications such as measurements, calculations, and comparisons. Irrational numbers are necessary because they represent quantities that cannot be expressed as fractions, such as the square root of non-perfect squares or transcendental numbers like π and e. Together, rational and irrational numbers form the real number system, which is fundamental for describing quantities in mathematics and the sciences.