As you guys know YouTube Short Videos are trending these days and many educational channels are posting short videos, we are planning to make a list of youtube short videos on topics of mathematics. This will help many YouTubers to create content on math topics. Here we go.

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Example 1:  Prove that $x^0=1$, that is, any number to the power zero is one.
Solution:
We have $x^0=x^{a-a}$
$=x^a \cdot x^{-a}$
$=x^a \cdot \frac{1}{x^a}$
$=1$.

Example 2:  Prove that $0! = 1$, that is, zero factorial is one.
Solution:
Note that $n! = n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1$
$\therefore n! = n \cdot (n-1)!$
$\Rightarrow (n-1)! = \frac{n!}{n}$
Put $n=1$
$\therefore 0! = \frac{1!}{1}=\frac{1}{1}=1$ (proved)

Example 3:  Why $0.999…=1$? (the value of  $0.999…$ is $1$)
Solution:

Example 4:  Prove that the natural logarithm of  $e$ is $1$, that is, $\ln(e)=1$.
Solution:
Let  $x=\ln(e)$
We know that the base of the natural \logarithm is $e$.
So we have  $x=\log_e(e)$
$\Rightarrow e^x=e \quad$ $[ \because x=\log_a b \Rightarrow a^x=b]$
$\Rightarrow e^x=e^1$
$\Rightarrow x=1. \quad$ So $x=\ln(e)=1$.

Example 5:  Calculate the natural logarithm of  $-1$? (Natural logarithm of a negative number)
Solution:
Let  $x=\ln(-1)$
We know that the base of the natural logarithm is $e$.
Also note that $-1=e^{i \pi}$
$\therefore x=\ln (-1)=\log_e(-1)$
$\Rightarrow x=\log_e e^{i \pi}$
$\Rightarrow x=i \pi \quad$ $[ \because \log_a a^b=b]$
So $x=\ln(-1)=i \pi$.

Example 6:  Calculate the natural logarithm of  $i$? (Natural logarithm of an imaginary complex number)
Solution:
Let  $x=\ln(i)$
We know that the base of the natural logarithm is $e$.
So we have  $x=\log_e(i)$
$\Rightarrow x=\log_e(\sqrt{-1})$
$\Rightarrow x=\log_e((-1)^{\frac{1}{2}})$
$\Rightarrow x=\frac{1}{2} \log_e(-1) \quad$ $[ \because \log_a x^b=b\log_a x]$
$\Rightarrow x=\frac{1}{2} \ln(-1)$.
So $x=\frac{1}{2} \cdot i \pi \quad$ $[\because \ln(-1)=i \pi]$ (see Example 5)
$\therefore x=\ln(i)=\frac{i \pi}{2}$

Example 7:  Prove that the derivative of a constant is 0.
Solution:
Let $f(x)=c$ be a constant function.
We will show that  $\frac{d}{dx}(f(x))=0$.
By definition, we have
$\frac{d}{dx}(f(x))=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$
$=\lim\limits_{h \to 0} \frac{c-c}{f}$
$=\lim\limits_{h \to 0} \frac{0}{h}=0 \quad$ (proved)

Example 8:  Find the sum of the first 100 natural numbers. Find 1+2+3+ …+100=?
Solution:
We know that $1+2+ \cdots +n=\frac{n(n+1)}{2}$
Put $n=100$
$\therefore 1+2+ \cdots +100=\frac{100 \times 101}{2}=5050$

Example 9:  Which one is larger? $\frac{2}{3}$ Vs $\frac{3}{4}$ (No Calculator)
Solution:
Aim: make both the denominators equal.
Note the denomina\tors of  $\frac{2}{3}$ and  $\frac{3}{4}$ are $3$ and $4$.
gcd (greatest common divisor) of  $3$ and $4$ is $12$.
We will make both the denominators 12 as follows:
$\frac{2}{3}$ Vs  $\frac{3}{4}$ $=\frac{2\times 4}{3 \times 4}$ Vs  $\frac{3 \times 3}{4 \times 3}$
$=\frac{8}{12}$ Vs  $\frac{9}{12}$
Since $9>8$ and we have the same denominators, we conclude that
$\frac{3}{4}> \frac{2}{3}$.