In mathematics, a number that is real can be defined as a finite real number, an infinite decimal extension. In other words, the real number cannot be used to solve a mathematical equation. On the other hand, real numbers are used only in measures of constantly changing numbers such as time and mass, compared to the natural number 1, 2, and the irrational number. In simple terms, real numbers cannot be changed without changing the whole universe. For example, if you are cooking dinner, and someone asks you how old your grandmother was, you could say “She was born on New Year’s Eve 16 96”.
However, by asking you how old your grandmother was, you have already used the finite real numbers, and thus, cannot answer the question with the infinite rational numbers. You cannot define the real numbers as “completeness” in mathematics. The real numbers do not have a complete solution, as stated above. Their existence as completeness in mathematics is strictly speaking not proven. For instance, the square root of -1, which is a prime number believed to exist as the real number representation of the real numbers, has no proven solutions. It was calculated by a person who found out the formula for it, but the exact results remain a mystery.
On the other hand, you can use the natural numbers as the perfect measure of completeness in mathematics. For instance, you can find the real numbers on the number line through a little process called counting the Fibonacci numbers. This “natural number line” will help you find the real solution for any problem. These solutions, however, are only obtainable when one is dealing with real numbers, and not the infinite irrational numbers or the real numbers themselves. Thus, theorems like “the real is greater than the ideal” and “the real is less than the ideal” only show the inaccuracy of our mathematics classes when we use the infinite irrational numbers as the measurement of the real numbers.
So, it is safe to say that both the real numbers and the natural numbers cannot be used as the measurement of the real numbers themselves. But, the accuracy of the measurement is only equivocal between the real numbers and the rational numbers. It is not as precise as the measurements between the real and irrational numbers. It eventually comes down to choosing the rational numbers that satisfy your measurement.
If you want to prove this inaccuracy of the whole numbers measurement, all you need to do is take the Fibonacci numbers as the measurement of the entire real number set. Now, if you put these Fibonacci numbers on a graph, you would immediately see that the whole numbers look more like points than actual real numbers on the graph. And this is why people say that the Fibonacci numbers represent the real numbers really well. If you were to put all the bigger numbers on a chart, you would eventually come to the conclusion that the bigger the numbers, the lesser their slope will be towards the end point of the graph. This means that the slope of the whole numbers will not eventually become zero.
Now, if you were to graph the real numbers or the irrational numbers separately, then it would seem that the slope of the whole numbers is just the same as the slope of the irrational numbers; and indeed it is. Their shapes are almost identical. And this similarity can also be seen between the natural numbers and the rational numbers as well. So, it must now be concluded that the whole numbers cannot truly be used as the measure of the real numbers.
If you want to calculate the value of the real number using the irrational numbers, then it must be assumed that the non-zero real number will eventually be divided by the greater irrational number and will then become the fraction of the greater irrational number. But, since the irrational number has no positive or negative sign, then it will always become a non-zero real number. Therefore, the calculation results that are derived using the real numbers will always be the sum of the fractions that exist between the natural and the irrational number. It is just a question of confirming if the whole number is less than the denominator.
The above discussion proves beyond doubt that the relationship between the real numbers and the irrational numbers does not support any claims of completeness in the study of numbers and their completeness. For the same reason, these results do not provide any guarantees of their accuracy. As the experts have already pointed out, the relationship between the real numbers and the completeness of mathematics has been disputed by many eminent scholars over the past few decades. For this reason, most experts feel that the debate should not be left to such experts. This is because the arguments of the experts should be more compelling than those of students who only study numbers for fun.