Let assume that $\sqrt7$ is a rational number. Therefore it can be expressed as a fraction $\dfrac{a}{b}$ where $a,b\in\mathbb{Z}$ and $\text{gcd}(a,b)=1$ .

$\sqrt7=\dfrac{a}{b}\\\\7=\dfrac{a^2}{b^2}\\\\a^2=7b^2$

This means that $a^2$ is divisible by 7, and therefore also $a$ is divisible by 7.

So, $a=7k$ where $k\in\mathbb{Z}$

$(7k)^2=7b^2\\\\49k^2=7b^2\\\\7k^2=b^2$

Analogically to $a^2=7b^2$ ------- $b^2$ is divisible by 7 and therefore so is $b$ .

But if both numbers $a$ and $b$ are divisible by 7, then $\text{gcd}(a,b)=7$ which contradicts our earlier assumption that $\text{gcd}(a,b)=1$ .

Therefore $\sqrt7$ is an irrational number.

8 0

4 years ago

Why the order of the coordinates does not matter when calculating length

Rudiy27

The formula for calculating length is:
$\sqrt{ (x_b-x_a)^{2} + (y_b-y_a)^{2} }$

We can also write $x_a - x_b$ or $y_a - y_b$
Why it does not matter?
Let's assume we have 2 numbers, a and b.
When we perform a subtraction:
$a-b$ , we get another number $c$
When we perform another subtraction:
$b-a$ , we get a number $-c$

When we raise $c$ or $-c$ to the power of 2, the result is the same, $c^{2}$ .

4 0

3 years ago