Is real number system combination of rational and irrational numbers?

Callum Dudley
2022-07-09
Answered

Is real number system combination of rational and irrational numbers?

You can still ask an expert for help

Freddy Doyle

Answered 2022-07-10
Author has **20** answers

Yes, that covers all of them, by definition.

We take the real numbers $\mathbb{R}$ and the rational numbers $\mathbb{Q}$. We note that $\mathbb{Q}$ is a proper subset of $\mathbb{R}$, so we take all the real numbers that aren't rational and call them irrational numbers, i.e. we define$\mathbb{I}=\mathbb{R}\setminus \mathbb{Q}$.

There are other splits possible. In particular, we can take all the real numbers that are roots of polynomials with integer coefficients, and call them the algebraic numbers (sometimes notated as $\mathbb{A}$, although that also sometimes includes the complex algebraic numbers). So we can also consider the set of what's left, and that's what we call the transcendental numbers $\mathbb{R}\setminus \mathbb{A}$. Since any rational number $\frac{p}{q}$ is the solution to $qx-p=0$, we can soon confirm that $\mathbb{Q}\subset \mathbb{A}$, i.e. the rational numbers form a proper subset of the algebraic numbers, and conversely the transcendental numbers form a proper subset of the irrational numbers.

We take the real numbers $\mathbb{R}$ and the rational numbers $\mathbb{Q}$. We note that $\mathbb{Q}$ is a proper subset of $\mathbb{R}$, so we take all the real numbers that aren't rational and call them irrational numbers, i.e. we define$\mathbb{I}=\mathbb{R}\setminus \mathbb{Q}$.

There are other splits possible. In particular, we can take all the real numbers that are roots of polynomials with integer coefficients, and call them the algebraic numbers (sometimes notated as $\mathbb{A}$, although that also sometimes includes the complex algebraic numbers). So we can also consider the set of what's left, and that's what we call the transcendental numbers $\mathbb{R}\setminus \mathbb{A}$. Since any rational number $\frac{p}{q}$ is the solution to $qx-p=0$, we can soon confirm that $\mathbb{Q}\subset \mathbb{A}$, i.e. the rational numbers form a proper subset of the algebraic numbers, and conversely the transcendental numbers form a proper subset of the irrational numbers.

asked 2022-11-06

How to/Can we show irrational numbers?

asked 2022-01-06

Can an irrational number raised to an irrational power be rational?

If it can be rational, how can one prove it?

If it can be rational, how can one prove it?

asked 2021-02-21

Which statement is false?

A. every irrational number is also a real number.

B. every integer is also a real number.

C. no irrational number is irrational.

D. every integer is also an irrational number.

A. every irrational number is also a real number.

B. every integer is also a real number.

C. no irrational number is irrational.

D. every integer is also an irrational number.

asked 2021-01-06

Write two rational and three irrational numbers that are between 3 and 4 with explanation.

asked 2022-06-12

Find a sequence of rationals that converges to arbitrary irrational number $\beta $.

asked 2022-05-24

How to find an irrational number in $\mathbb{Q}\cap [0,1]$?

asked 2022-08-02

What is an irrational number that is not a real number?