Skip to main content

Why can't you divide by zero?

In the world of math, many strange results are possible when we change the rules. But there’s one rule that most of us have been warned not to break: don’t divide by zero. How can the simple combination of an everyday number and a basic operation cause such problem?                                     

Hands holding the number 0245990083

Normally, dividing by smaller and smaller numbers gives you bigger and bigger answers. Ten divided by two is five, by one is ten, by one-millionth is 10 million, and so on. So it seems like if you divide by numbers that keep shrinking all the way down to zero, the answer will grow to the largest thing possible. Then, isn’t the answer to 10 divided by zero actually infinity? That may sound plausible. But all we really know is that if we divide 10 by a number that tends towards zero, the answer tends towards infinity. And that’s not the same thing assaying that 10 divided by zero is equal to infinity. Why not? Well, let’s take a closer look at what division really means. Ten divided by two could mean, "How many times must we add two together to make 10,” or, “two times what equals 10?” Dividing by a number is essentially the reverse of multiplying by it, in the following way: if we multiply any number by a given number x, we can ask if there’s a new number we can multiply by afterwards to get back to where we started. If there is, the new number is called the multiplicative inverse of x. For example, if you multiply three by two to get six, you can then multiply by one-half to get back to three. 

So the multiplicative inverse of two is one-half, and the multiplicative inverse of 10 is one-tenth. As you might notice, the product of any number and its multiplicative inverse is always one. If we want to divide by zero, we need to find its multiplicative inverse, which should be one over zero. This would have to be such a number that multiplying it by zero would give one. But because anything multiplied by zero is still zero, such a number is impossible, so zero has no multiplicative inverse. Does that really settle things, though? 

After all, mathematicians have broken rules before. For example, for a long time, there was no such thing as taking the square root of negative numbers. But then mathematicians defined the square root of negative one as a new number called I, opening up a whole new mathematical world of complex numbers. So if they can do that, couldn’t we just make up a new rule, say, that the symbol infinity means one over zero, and see what happens? Let's try it, imagining we don’t know anything about infinity already. Based on the definition of a multiplicative inverse, zero times infinity must be equal to one. 

That means zero times infinity plus zero times infinity should equal two. Now, by the distributive property, the left side of the equation can be rearranged to zero plus zero times infinity. And since zero plus zero is definitely zero, that reduces down to zero times infinity. Unfortunately, we’ve already defined this as equal to one, while the other side of the equation is still telling us it’s equal to two. So, one equals two. Oddly enough, that's not necessarily wrong; it's just not true in our normal world of numbers. There’s still a way it could be mathematically valid, if one, two, and every other number were equal to zero. But having infinity equal to zero is ultimately not all that useful to mathematicians, or anyone else.

 There actually is something called the Riemann sphere that involves dividing by zero by a different method, but that’s a story for another day. In the meantime, dividing by zero in the most obvious way doesn’t work out so great. But that shouldn’t stop us from living dangerously and experimenting with breaking mathematical rules to see if we can invent fun, new worlds to explore. 

Thank you

Comments

Post a Comment

Popular posts from this blog

Savitri and Satyavan: The legend of the princess who outwitted Death

Princess Savitri of the Madra Kingdom was as benevolent, brilliant, and bright as the Sun God she was named for. Her grace was known throughout the land, and many powerful princes and wealthy merchants flocked to her family’s palace to seek her hand in marriage.  But upon witnessing her blinding splendor in person, the men lost their nerve. Unimpressed with these suitors, the princess determined to find a husband herself. Mounting her golden chariot, she travelled through rolling deserts, glittering cities and snow-capped mountains— rejecting many men along the way. Eventually, Savitri ventured deep into the jungle, where she met a young man chopping wood. His name was Satyavan, and like her, he loved the tranquil forest— but the princess saw he was not at peace. After talking for hours, Satyavan told her of his plight. His parents had once been wealthy rulers, until his father was blinded and overthrown in a violent coup.  Now Satyavan worked tirelessly to support their meag...
Post a Comment
Read more

Santizer VS Soap / Which is best for Corona ?

Your hands, up close, are anything but smooth. With peaks and valleys, folds and rifts, there are plenty of hiding places for a virus to stick. If you then touch your face, the virus can infect you. But there are two extraordinarily simple ways you can keep that from happening: soap and water, and hand sanitizer. So which is better?  The coronavirus that causes COVID-19 is one of many viruses whose protective outer surface is made of a lipid bilayer. These lipids are pin shaped molecules whose heads are attracted to water, and tails are repulsed by it. So in water-rich environments, lipids naturally form a shell like this, with the heads outside and the tails inside. Their shared reaction to water makes the lipids stick loosely together— this is called the hydrophobic effect. This outer structure helps the molecular machinery of the virus break through cellular membranes and hijack our cells. But it has thousands upon thousands of weak points where the right molecules could pr...
Post a Comment
Read more

How to make your writing suspenseful ?

What makes a story good? Sure, you could throw in some hideous monsters, fountains of blood, and things jumping out from every corner, but as classic horror author H.P. Lovecraft wrote, "The oldest and strongest kind of fear is fear of the unknown." And writers harness that fear not by revealing horrors, but by leaving the audience hanging in anticipation of them. That is, in a state of suspense.   The most familiar examples of suspense come from horror films and mystery novels. What's inside the haunted mansion? Which of the dinner guests is the murderer? But suspense exists beyond these genres. Will the hero save the day? Will the couple get together in the end? And what is the dark secret that causes the main character so much pain? The key to suspense is that it sets up a question, or several, that the audience hopes to get an answer to and delays that answer while maintaining their interest and keeping them guessing. So what are some techniques you can use to achieve...
Post a Comment
Read more