Step into the fascinating world of mathematics and discover the hidden gems that lie within rational numbers. These little wonders are the building blocks of all mathematical concepts, from the basic to the advanced. They are the key to unlocking the mysteries of numerical understanding and unleashing the full potential of mathematical applications. We will delve deep into the essence of rational numbers, uncovering their properties, examples, and practical implications. Let us show you how rational numbers can pave the way to mathematical success.
What Are Rational Numbers?
The concept of rational numbers is fundamental to mathematics and plays a crucial role in the number system. This section aims to provide an exploration of rational numbers and their significance in mathematics. Rational numbers are defined as numbers that can be expressed as the quotient or fraction of two integers. Here the numerator is an integer and the denominator is a non-zero integer. Mathematically, a rational number ‘q’ can be represented as $$\frac{p}{q},$$ where ‘$p$’ and ‘$q$’ are integers. Note that the denominator ‘$q$’ cannot be zero, as division by zero is undefined in mathematics.
The rational numbers are represented by the symbol $\mathbb{Q}$. An element $x$ belonging to $\mathbb{Q}$ is expressed as $x=\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not equal to zero. This notation signifies that the set of rational numbers is composed of fractions with integer numerators and denominators. In fact, the use of this notation is essential in the study of mathematical concepts such as algebraic structures, number theory, and analysis. The representation of rational numbers in this manner provides a concise and precise way of expressing mathematical ideas.
Properties of a rational number
The present study aims to demonstrate that every rational number can be expressed in a simplified form. To illustrate this property, consider the example of $\frac{4}{18}$, which can be simplified as $\frac{2}{9}$ by dividing both the numerator and denominator by their greatest common factor of $2$. This observation leads to the conclusion that any rational number can be represented as an integer or a fraction of the form $\frac{p}{q}$, where $p$ and $q$ are integers with a greatest common divisor of $1$. This result has significant implications in various mathematical fields, including algebra and number theory.
| If $x,y\in\mathbb{Q}$, then $xy\in\mathbb{Q}$ and $x+y\in\mathbb{Q}$. | + |
| Given $x,y\in\mathbb{Q}$, there exist integers $a,b,c,d$ such that $b\neq 0$, $d\neq 0$, $x=\frac{a}{b}$, and $y=\frac{c}{d}$. Consequently, $x+y=\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$, which implies that $x+y$ is a rational number. Similarly, $xy=\frac{ac}{bd}$, which is also a rational number. | |
Irrational numbers
Irrational numbers are a subset of real numbers that cannot be expressed as a ratio of two integers. They are characterized by their decimal representations, which neither terminate nor repeat in a regular pattern. Examples of irrational numbers include the square root of 2 and the mathematical constant pi. Unlike rational numbers, irrational numbers possess an infinite and non-repeating decimal expansion. These numbers are fundamental in various mathematical contexts, particularly in geometry and calculus, where they appear in equations involving circles, curves, and right triangles. A thorough understanding of irrational numbers enhances our comprehension of the complex nature of real numbers and their applications in mathematics and science.
| Exercise: Prove that $\sqrt{2}\notin \mathbb{Q}$. This $\sqrt{2}$ is irrational. | + |
| Let us assume, by way of contradiction, that $\sqrt{2}\in \mathbb{Q}$. This implies the existence of two integers of the same sign, $p$ and $q$, with $q\neq 0$ and the greatest common divisor of $p$ and $q$ being $1$, such that $\sqrt {2}=\frac{p}{q}$. Without loss of generality, we may assume that $p$ and $q$ are positive. Squaring both sides of this equation yields $p^2=2 q^2$. This implies that $p^2$ is even, and therefore $p$ is even. Thus, there exists an integer $k\in\mathbb{N}$ such that $p=2k$. Substituting $p$ with its value, we obtain $4k^2=2 q^2$. Hence, $q^2=2k^2$, which implies that $q^2$ is even. Therefore, $q$ is even. We have thus shown that both $p$ and $q$ are even, which contradicts the fact that $p$ and $q$ are coprime. | |
Operations on irrational numbers
| The addition of a rational number and an irrational number results in an irrational number. | + |
| Let us consider a rational number, denoted by $r$, and an irrational number, denoted by $w$. If the sum of $r$ and $w$ is rational, denoted by $r’$, then it follows that $w=r’-r$ is also rational. However, this assertion is untenable. | |
| The sum of two irrational numbers is SOMETIMES irrational. | + |
| The number $\sqrt{2}+\sqrt{3}$ is proven to be irrational through a proof by contradiction. Suppose there exists a rational number $r$ such that $\sqrt{2}+\sqrt{3}=r$. It follows that $5+2\sqrt{6}=r^2$. As $r^2$ is rational, $r^2-5$ is also rational. Therefore, $\frac{r^2-5}{2}$ is rational, which implies that $\sqrt{6}$ is rational. Consequently, $\sqrt{6}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ have a greatest common divisor of $1$. Since $p^2=6q^2$, it follows that $p$ is even. From $4k^2=2\cdot6q^2$, it can be deduced that $2k^2=3q^2$, which shows that $q^2$ is even. Thus, $q$ is even. This contradicts the assumption that $p$ and $q$ have a greatest common divisor of $1$. Therefore, $\sqrt{2}+\sqrt{3}$ is irrational. | |
| The multiplication of rational and irrational numbers results in an irrational number. | + |
| In fact, consider a rational number denoted by $r$ and an irrational number denoted by $w$. Supposing that their product, $rw$, yields a rational number, denoted by $r’$, it follows that $w$ can be expressed as the ratio of two integers, namely $\frac{r’}{r}=\frac{p}{q}$. This assertion, however, is untenable.. | |
