# Division by zero & what is infinity?

You can’t divide a number by 0” how many times have you heard this? Don’t you think that even the seemingly impossible can actually be possible? Let’s think of zero as a really small number, like 0.0000000000001; we are going to divide by a much bigger number, like 100, until we get to our new “definition” of zero. Let’s first see what 1 divided by 0 is equal to by using the approach I have just mentioned.

1/100=0.01

1/50=0.02

1/10=0.1

1/5=0.2

1/1=1

1/0.5=2

1/0.1=10

1/0.01=100

1/0.001=1000

1/0.0001=10000

1/0.00001=100000

You can clearly see that the smaller the number we are dividing by, the greater the result. So, if you divide by 0.0000000000001 you’ll get 1000000000000, but zero is much smaller than 0.0000000000001, which means you’ll get an incredibly big number. This number is called infinity. In general:

$\frac{a}{0}=\infty$

with a > 0.

To be more correct though, we must write infinity as the limit of a/n where n tends to 0 (remember that a is a non-zero number).

$\lim_{n\to&space;0}\,&space;\frac{a}{n}=\infty&space;\,&space;\,&space;\,&space;$

with a > 0; if a is negative, we can write it as (-1)|a|, where |a| is positive. Therefore

$\lim_{n\to&space;0}\,&space;(-1)\frac{|a|}{n}\rightarrow&space;(-1)\lim_{n\to&space;0}&space;\frac{|a|}{n}=-\infty$

Positive nfinity and negative infinity are such large numbers that adding or subtracting a very big number like 10000000000000000000000000000000 won’t change a thing.

$\infty&space;\,&space;\cdot&space;\,&space;5000=\infty$ $-&space;\infty&space;\,&space;\cdot&space;\,&space;100000000=-\infty$ $\infty&space;\,&space;\cdot&space;\,&space;-900000000=-\infty$ $-\infty&space;\,&space;\cdot&space;\,&space;-900000000=+\infty$ $-\infty&space;\,&space;:&space;\,&space;\left&space;(&space;-10^{20}&space;\right&space;)=+\infty$ $0\,&space;:&space;\,&space;\infty&space;=&space;0$ $\infty&space;\,&space;-&space;\,&space;15^{74}=\infty$ $-\infty&space;\,&space;+&space;\,&space;90^{100}=\infty$ $\infty^2=\infty$ $\sqrt\infty=\infty$

But what about 1/infinity? Let’s use the same approach we used for 1/0, but in reverse: let’s say infinity is 10000000000000000000000. We will start by dividing by a small number and then proceed by increasing it.

1/1=1

1/10=0.1

1/100=0.01

1/1000=0.001

1/10000=0.0001

1/100000=0.00001

You can see that the exact opposite is happening here! As the number we divide by increases, the result decreases and approaches 0.

$\lim_{n\to\infty}\,&space;\frac{a}{n}=0$

In another post I will explain what infinity/infinity and 0/0 give, but to do that I will have to explain limits in a more detailed way, so wait for it! In the meantime, you can try to figure it out and leave the answer in the comments! Subscribe and stay tuned for more!

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