High School Number Theory Archives - Page 8 of 11

Proof that log 2 is an irrational number

January 6, 2012 GB High School Mathematics, High School Number Theory

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.

One comment irrational numbers, log 2, proof by contradiction, rational numbers

The 12 Days of Christmaths and the Triangular Numbers

December 24, 2011 GB High School Mathematics, High School Number Theory

Many of us are familiar with the Twelve Days of Christmas — a song that enumerates a series of increasingly grand gifts given on each of the twelve days of Christmas. For those who are not familiar with the song, here are a few stanzas (for full lyrics, click here).

On the first day of Christmas,
my true love sent to me
A partridge in a pear tree.

On the second day of Christmas,
my true love sent to me
Two turtle doves,
And a partridge in a pear tree.

On the third day of Christmas,
my true love sent to me
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the fourth day of Christmas,
my true love sent to me
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

The math: How many gifts was given on the twelfth day? » Read more

2 comments 12 days of christmas, rectangular numbers, triangular numbers

Divisibility by 4

November 10, 2011 GB High School Mathematics, High School Number Theory

This is the second post in the Divisibility Rules Series. In the last post, we discussed about divisibility by 2. In this post, we discuss divisibility by 4.

Now, how do we know if a number is divisible by $4$ ?

Four divides $100$ because $4 \times 25 = 100$ . It is also clear that four divides $200, 300, 400$ and all multiples of $100$ . Therefore, four divides multiples of $1000$ , $10 000$ , and $100 000$ . In general, $4$ divides $10^n$ , where $n$ is an integer greater than $1$ .

Now, how do we know if a number that is not a power of $10$ is divisible by $4$ . Let us try a few examples.

Example 1: Is $148$ divisible by $4$ ? $148$ is equal to $100 + 48$ and $100$ is divisible by $4$ . Since $48$ is also divisible by $4$ , therefore, $148$ id divisible by $4$ .

Example 2: Is $362$ divisible by $4$ ? $362$ is equal to $300 + 62$ . Now, $300$ is divisible by $4$ . Since $62$ is not divisible by $4$ , therefore, $362$ is not divisible by $4$ .

Example 3: Is $3426$ divisible by $4$ ? $3426 = 3400 + 26$ . Now, $3400$ is divisible by $4$ (it’s a multiple of 100), and $26$ is not divisible by $4$ . Therefore, $3426$ is not divisible by $4$ .

By now, you would have realized that we just test the last 2 digits of the numbers if we want to find out if it is divisible by 4: 148, 362, and 3426. » Read more

4 comments divisibility, divisibility rules, divisibility test, divisiblity by 4

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