YouTube Shorts on Mathematics
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.
First, we learn some benefits of uploading youtube shorts.
Benefits of YouTube Shorts:
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
$\frac{3}{4}> \frac{2}{3}$.