There are some rules in mathematics that cannot be changed from the past to the present. One of them is that you can never divide a number by zero. We are with you in our article where we will answer the question of whether the number is infinite or undefined when divided by zero.
maths It is an interesting branch of science in itself, because it sheds light on nature’s most ancient mysteries. Mathematics, which is difficult for some and a very fun course for others. Zero also has a special place. there is. Because one of the most memorable mathematical rules is that you cannot divide a number by zero.
In real numbers, that is, in the set of all numbers on the number line, a number division by zero we were taught in schools as undefined. Well, have you ever wondered why if you divide a number by zero, the result is undefined? Let’s look at the answer to this question together.
Why is a number not divisible by 0?
Any real number divided by zero is undefined. Division by zero is not defined in the division operation. It is also not infinite. The result of the division can be positive infinity or negative infinity. Neither infinity denotes a real number, so we can say with certainty that number over zero is undefined.
For example, to think about what you get when you divide the number 10 by zero; Let’s start by dividing 10 by 5. This process answer it will be 2. What if you divided 10 by a smaller number, 2? More a big number you would get the 5. What about 10 dividing by 1?
Again, a larger number emerges. 10. 10 divided by ½ is 20. 40 when divided by ¼; It is 320 when divided by 1/32. When to a smaller number If you divide, you get a larger number in return. That is, the closer the divisor gets to 0; the closer your answer is to infinity. So if you actually divided 10 by 0, you would get infinity, right?
In order to get a result in this process You can only get a limit. (Number / X), as x goes to zero, and you can examine the behavior of this function. However, we do not encounter a single limit value here either, because the limits are different from right and left. When approaching zero from the right, the value of the function goes positively towards infinity. It is an infinite abbreviation, meaning that the result is greater than any real number X, constantly growing, and therefore never certain. All you can say is that you can get a large result from any real number you want.
Approaching zero from the left If so, the situation is the same in the negative direction. The number becomes very large in absolute value but is negative, so shrinks. It’s a smaller number than any negative real number you can specify, so it’s negative. is infinite is called. As we said, X/0 has no value, a defined process is not. You can never divide by zero.
Infinity is already in the set of real numbers. a defined number since it is not strictly defined. Zero divided by zero, one to the infinity, infinity over infinity, infinity minus infinity are uncertainties. It is not undefined. In order to do these operations, we can eliminate the situation that causes uncertainty and find the result.
division by zero, you can’t find the answer is an operation, so the result of the operation is undefined. You can understand why if we look at the relationship between division and multiplication. If you divide 6 by 3, the answer is 2, because 2 times 3 = 6. If you divide 6 by zero, “What number gives zero times 6?” You ask the question. The answer to this, of course, is not a number, because we know that any real number zero times zero is not 6. For this reason divide by zero We say it is undefined because division by other numbers is not consistent.
Think about, to infinity in all directions There is a two-dimensional plane that goes out and has no center in the middle. Now that you bend this plane and turn it into a sphere and zero is the south pole, the corners are at the very top; Imagine it converging at the north pole. Now, take another infinite two-dimensional plane and place it crossing the equator. selected on this plane. any point, can be connected to the North Pole of the sphere by a straight line. If the point you selected is outside the sphere; the connecting line will intersect the globe with the northern hemisphere. If it is within the sphere, it will intersect in the southern hemisphere.
What you dream of is a Riemann Sphere. This method involves associating every point on the plane with an intersection point on the sphere. stereographic projection it’s called. Basically, any point you can find on the plane can be found on the sphere. This includes eternity. The closer you get to infinity on the plane, the closer you get to the sphere’s North Pole.
If we explain with another example;
What happens if you add apples to oranges? Of course it doesn’t make sense, so the easiest thing is that it doesn’t make sense, or as one mathematician said, “undefined” to say it is. Maybe that’s the best way to look at it. in mathematics, “Operation XYZ is undefined” When you see an expression like “Operation XYZ doesn’t make sense” You can think of it as
Another way to think about it is to imagine filling a box with apples. Suppose there are 100 apples in a box. Now try filling it with apples half the size of those apples. You can put 200 apples in the box. Now special that takes up no space, a magic apple imagine. How many can you put in the box?
This process any answer none. Therefore, mathematicians use numbers divisible by 0. “undefined” calls it. Some researchers tend to view this process as infinite, but this process is not entirely accurate. First of all, dividing a number by zero can be thought of as infinity at first glance. Because the smaller the divisor, the larger the result. For example, if we divide the number 10 into smaller numbers at each step, we see that the result gets bigger.
- 10 / Divider = Result
- 10 / 1 = 10
- 10 / 0.1 = 100
- 10 / 0.01 = 1000
- 10 / 0.001 = 10000
- 10 / 0.0001 = 100000
- 10 / 0.00001 = 1000000
- 10 / 0.000001 = 10000000
- Relation = 10/x = y
As you can see how many divisors if small the greater the result. In other words, the closer the divisor gets to zero, the closer the result is to infinity. So a number divided by zero must be infinity.
To understand why this is not true, we must first know what division means. For example, when we divide 10 by 2, the result is 5. This operation tells us how many 2s are in 10. Also, division is mathematically the opposite of multiplication. If we arrange the results of the division and multiplication operations to be equal to each other, the concept of multiplicative inverse emerges.
- 10 / 5 = 2 = 10 x 1/5
- 10/2 = 5 = 10 x 1/2
- 10/a = 10 x 1/a
The number 1/a in operations is called the multiplicative inverse. In the first operation, the multiplicative inverse of 5 is 1/5, in the second operation, the multiplicative inverse of 2 is 1/2. So the multiplicative inverse of a number is 1 divided by that number. (the multiplicative inverse of a is 1/a). So what good will the multiplicative inverse do? Multiplying a number by its multiplicative inverse always yields 1.
- Number x multiplicative inverse = 1
- 5 x 1/5 =1
- 2 x 1/2 = 1
- 4000 x 1/4000 = 1
- ax 1/a = 1
In this case, the multiplicative inverse of zero should be 1/0, and the multiplicative inverse of zero should also give 1. (0 x 1/0 = 1). Here the problem arises. Because zero multiplied by a number is zero. Therefore, zero does not have a multiplicative inverse. So 1/0 is undefined. If we write the division of numbers by zero as a product,
- 5/0 = 5 x 1/0
- 10/0 = 10 x 1/0
- -3/0 = -3 x 1/0
Since 1/0 is undefined, all results are undefined. So a number divided by zero is undefined. The one I mentioned in the first part and which seems logical 1/0 = Infinite The idea may still sound logical. But this gives the same result for negative numbers. When we divide a number by negative numbers approaching zero, the result approaches minus infinity.
- 10 / Negative Divider Number = Result
- 10 / -1 = -10
- 10 / -0.1 = -100
- 10 / -0.01 = -1000
- 10 / -0.001 = -10000
- 10 / -0.0001 = -100000
- 10 /- 0.00001 = -1000000
- 10 /- 0.000001 = -10000000
- Relation = 10/-x = -y
Therefore, if the result of 1/0 is infinity, the result of 1/-0 must also be negative infinity. Since zero is a neutral number, both plus infinity and minus infinity results for 1/0. Since plus infinity and minus infinity are not equal to each other, this thought is wrong it appears to be.
If all these processes confuse you simple logic Let’s go over. As I mentioned in the previous sections, for example, 10/5 shows how many 5s are in 10. Accordingly, the 10/0 operation should show how many zeros are in 10. How many zeros are in 10? 1, 10, infinity… we don’t know because the result is not mathematically defined. Therefore, if we divide a number by zero, the result is undefined.
In this article, we tried to explain why a number cannot be divided by zero with all the details. In this article, which we think is useful for those who are curious, what happens when we divide a number by zero, the result Is it undefined or infinite? We sought answers to your questions. Stay safe until we meet in our next article.