Tag Archives: rational numbers

Are all fractions rational numbers?

Posted on January 6, 2013 by Guillermo Bautista | 3 Comments

No.

A rational number can be expressed in the form $\displaystyle\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ . In other words, it is a fraction whose denominator is not zero, and both the denominator and numerator are integers.

$fractions$

Some fractions, however, may contain a numerator or denominator that is not an integer. Some examples of such fractions are

$\displaystyle\frac{\sqrt{3}}{2}$ , $\displaystyle\frac{\pi}{4}$ and $\displaystyle\frac{e}{2}$ .

A rational number may be represented in many ways, but it can always be expressed as a fraction. For instance, $10^{-1}$ is a rational number because we can express it as $\frac{1}{10}$ . Also, the number $0.333 \cdots$ , a repeating decimal, is a rational number because we can also express it as fraction $\frac{1}{3}$ .

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Posted in High School Mathematics, High School Number Theory, Question and Answer

Tagged are all rational fractions, area all fractions rational, fractions, rational number examples, rational numbers, repeating decimals

Why are Non-terminating, Repeating Decimals Rational

Posted on August 11, 2012 by Guillermo Bautista | 3 Comments

Last night, I received a Facebook message from a Grade 8 student asking why non-terminating repeating decimals are rational. I am posting the answer here for reference.

Rational numbers is closed under addition. That is, if we add two rational numbers, we are guaranteed that the sum is also a rational number. The proof of this is quite easy, so I leave it as an exercise for advanced high school students.

Before discussing non-terminating decimals, let me also note that terminating decimals are rational. I think this is quite obvious because terminating decimals can be converted to fractions (and fractions are rational). For example, $0.842$ can be expressed as

$\displaystyle\frac{842}{1000}$ .

Further, terminating decimals can be expressed as sum of fractions. For example, $0.842$ can be expressed as

$\frac{8}{10} + \frac{4}{100} + \frac{2}{1000}$ .

Since rational numbers is closed under addition, the sum of any number of fractions is also a fraction. This shows that all terminating decimals are fractions. Continue reading →

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Posted in High School Mathematics, Question and Answer

Tagged non-terminating repeating decimals, non-terminating repeating decimals to fractions, rational numbers, terminating decimals

Proof that log 2 is an irrational number

Posted on January 6, 2012 by Guillermo Bautista | 1 Comment

Before doing the proof, let us recall two things: (1) rational numbers are numbers that can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers, and $b$ not equal to $0$ ; and (2) for any positive real number $y$ , its logarithm to base $10$ is defined to be a number $x$ such that $10^x = y$ . In proving the statement, we use proof by contradiction.

Theorem: log 2 is irrational

Proof:

Assuming that log 2 is a rational number. Then it can be expressed as $\frac{a}{b}$ with $a$ and $b$ are positive integers (Why?). Then, the equation is equivalent to $2 = 10^{\frac{a}{b}}$ . Raising both sides of the equation to $b$ , we have $2^b = 10^a$ . This implies that $2^b = 2^a5^a$ . Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because $2^b$ is an integer that is not divisible by 5 for any $b$ , while $2^a5^a$ is divisible by 5. This means that log 2 cannot be expressed as $\frac{a}{b}$ and is therefore irrational which is what we want to show.