Is 70.300722 irrational or rational

Determining the Nature of 70.300722

To determine whether the number $70.300722$ is rational or irrational, we need to understand the properties of both types of numbers.

Rational Numbers

A number is considered rational if it can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. Formally, a rational number $q$ satisfies:

Irrational Numbers

A number is considered irrational if it cannot be expressed as a simple fraction. Irrational numbers have non-repeating, non-terminating decimal expansions.

Analyzing 70.300722

The number $70.300722$ has a finite decimal representation. It can be written as:

Here, $a = 70300722$ and $b = 1000000$, both of which are integers. This demonstrates that $70.300722$ can be expressed as the fraction of two integers.

Conclusion

Since $70.300722$ can be expressed in the form $\frac{a}{b}$ with $a$ and $b$ as integers and $b \neq 0$, it qualifies as a rational number.

Therefore, 70.300722 is a rational number.

Definition of Rational Numbers

A rational number is any number that can be expressed in the form of $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$. The set of rational numbers is denoted by $\mathbb{Q}$.

Properties of Rational Numbers

1. Closure Property

Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero). This means that performing these operations on rational numbers will always result in another rational number.

2. Commutative Property

Addition and multiplication of rational numbers are commutative, meaning the order in which you add or multiply two rational numbers does not affect the result.

3. Associative Property

Addition and multiplication of rational numbers are associative, meaning the grouping of numbers does not affect the result.

4. Distributive Property

Multiplication over addition for rational numbers is distributive. This means multiplying a number by a sum is the same as doing each multiplication separately and then adding.

5. Identity Property

The additive identity for rational numbers is 0, and the multiplicative identity is 1.

6. Inverse Property

Every rational number has an additive inverse (opposite) such that their sum is 0, and a multiplicative inverse (reciprocal) such that their product is 1 (except for 0).

Understanding rational number properties is fundamental in arithmetic and algebra, as they establish how these numbers interact within the number system.

Understanding Irrational Number Properties

Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. They have several unique properties that distinguish them from rational numbers.

Definition

An irrational number is defined as a number that cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Examples of Irrational Numbers

• Pi ($\pi$): Approximately equal to 3.14159.
• Euler's Number ($e$): Approximately equal to 2.71828.
• Square Roots of Non-Perfect Squares: For example, $\sqrt{2}$, $\sqrt{3}$, etc.

Properties

Non-Terminating and Non-Repeating Decimals

One of the key properties of irrational numbers is that when expressed in decimal form, they do not terminate and do not repeat. For instance,

$\pi = 3.141592653589793238 \ldots$

Closure Properties

• Addition: The sum of two irrational numbers can be either rational or irrational. For instance, $\pi + (-\pi) = 0$ (rational) but $\pi + e$ is irrational.

• Multiplication: The product of two irrational numbers can also be either rational or irrational. For example, $\pi \cdot \frac{1}{\pi} = 1$ (rational), but $\pi \cdot e$ is irrational.

Irrationality of Certain Operations

Certain operations involving irrational numbers retain their irrationality:

$\sqrt{2} + \sqrt{3} \neq \sqrt{5}$

Instead:

$\sqrt{2} + \sqrt{3}$

Proof of Irrationality by Contradiction

To prove a number is irrational, a common method is proof by contradiction. Let's look at the proof for $\sqrt{2}$:

Assume $\sqrt{2}$ is rational:

$\sqrt{2} = \frac{p}{q}$

where $p$ and $q$ are co-prime integers. Then,

This implies $p^2$ is even, hence $p$ must be even. Let $p = 2k$:

Similarly, $q^2$ must be even, implying $q$ is even. But this contradicts the assumption that $p$ and $q$ are co-prime. Hence, $\sqrt{2}$ is irrational.

Conclusion

Understanding the properties of irrational numbers provides insight into the broader field of real numbers and numerical analysis. Such knowledge is essential for advanced mathematical concepts and real-world applications.