# If 1 is equal to 0 9999999999999

## Right / right?: Is 0.99999999999999999 ... equal to 1?

This week's question does not come from an inquisitive reader - the mathematician and ZEIT author Andreas Loos drew my attention to an essay by Thomas Bedürftig from the University of Hanover that deals with this problem.

Mathematically educated people usually answer the question: Of course 0.999 is ... 1! If, on the other hand, you ask students in grades 7 to 12, as did the didactic Ludwig Bauer from the University of Passau, most of them answer: 0.999 ... is less than 1. There is an "infinitely small" difference between the two numbers. But infinitely small numbers do not exist in the mathematics of real numbers.

There are several ways to justify that the two numbers are the same. For example, with the fact that one third is known to be 0.333 ... and consequently three thirds is 0.999 ... and three thirds are of course equal to 1. Somewhat more mathematical: The three points mean that you are looking at a so-called limit value. With more and more places after the decimal point, one approaches 1 at will, the difference is smaller than any positive number, i.e. 0.

Should one turn up one's nose at the naivety of the students? There is actually an alternative mathematics in which there are also infinitely small values - the so-called hyper-real numbers. And there is 0.999 ... also less than 1. Leibniz, for example, calculated with such numbers when he invented infinitesimal calculus. So the correct answer to the question is: It depends on what kind of math you do.

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