90.790170 is it irrational or not

Understanding Rational and Irrational Numbers

To determine if a number like 90.790170 is irrational or not, we need to recall the definitions of rational and irrational numbers:

Rational Numbers

A rational number is any number that can be expressed as the fraction of two integers, where the denominator is not zero. In other words, a number $r$ is rational if there exist integers $p$ and $q \neq 0$ such that:

Irrational Numbers

An irrational number is a number that cannot be written as a simple fraction of two integers. These numbers have non-repeating, non-terminating decimal expansions.

Analysis of the Given Number

The number 90.790170 has a finite decimal representation. Since it stops after a certain number of decimal places, it can be expressed as a fraction of two integers.

Fraction Representation

To express 90.790170 as a fraction, consider the following conversion steps:

1. Write down the number without the decimal point:

$90790170$

2. Determine the place value of the last digit:

Since there are 6 digits after the decimal point, the number can be written as:

$90.790170 = \frac{90790170}{1000000}$

3. Simplify the fraction (if possible) to find the simplest form:

If we simplify, we find:

Thus, 90.790170 is a rational number.

Conclusion

Given that 90.790170 has a finite decimal representation and can be expressed as a fraction of two integers, it is undoubtedly a rational number.

Rational and irrational numbers

Rational numbers are numbers that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ and $q$ are integers and $q \neq 0$. These numbers can be written in decimal form, where the decimal either terminates or repeats. Examples of rational numbers include:

In decimal form, they might look like:

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. Their decimal form is non-terminating and non-repeating. Typical examples include:

Key Properties of Rational Numbers

1. Form: Can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
2. Decimal Representation: Either terminates or repeats.

Key Properties of Irrational Numbers

1. Form: Cannot be written as $\frac{a}{b}$, where $a$ and $b$ are integers.
2. Decimal Representation: Neither terminates nor repeats.

Examples with Formulas

For rational numbers, consider:

For irrational numbers:

Mixed Example

Not all square roots are irrational. For instance: $\sqrt{9} = 3$ is rational, whereas: $\sqrt{5} \approx 2.23606797749979 \ldots$ is irrational.

In conclusion, understanding rational and irrational numbers helps in identifying the nature of numbers and their decimal behavior. This distinction is foundational in various fields of mathematics and applied sciences.

Definition

A finite decimal representation of a number is a way of expressing that number using a finite string of digits after the decimal point. This implies that the number terminates and does not continue indefinitely.

Characteristics

To have a finite decimal representation, a number must meet specific conditions:

• It must be a rational number.
• When expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$, the denominator $q$ should only have the prime factors 2 and 5.

Mathematical Explanation

If a fraction $\frac{p}{q}$ has a finite decimal representation, the fraction can be written such that:

where $a$, $m$, and $n$ are integers, and $m$ and $n$ are non-negative.

For example:

• $\frac{1}{8} = \frac{1}{2^3}$, which equals $0.125 (finite decimal).

• $\frac{7}{50} = \frac{7}{2 \cdot 5^2}$, which equals $0.14$ (finite decimal).

Counterexamples

Numbers that do not have a finite decimal representation include:

• Numbers where the prime factorization of the denominator includes primes other than 2 or 5. For instance, $\frac{1}{3}$ results in the repeating decimal $0.\overline{3}$.
• Numbers that are irrational. For example, the decimal representation of $\pi$ is non-terminating and non-repeating.

Importance

Understanding finite decimal representation is important for:

• Exact arithmetic calculations in various fields such as engineering, finance, and computer science.
• Simplifying the representation of fractions and understanding their properties.

Key takeaway:

A number has a finite decimal representation if it can be expressed as a fraction where the denominator's only prime factors are 2 and 5.