Decimal development of the unit

The decimal development of the unit is a qualified mathematical curiosity of Paradoxe because of its against-intuitive character. It corresponds to the equality between the two writings of the decimal Développement of the unit:

1 = 0, \ underline {9} , with 0, \ underline {9} = 0,99999…

First demonstration (via solution of an equation)

One poses the variable X :

x = 0, \ underline {9}

While multiplying by 10, it follows that:

10x = 9, \ underline {9}

One proceeds to a subtraction between the two preceding equations:

9x = 9, \ underline {9} - 0, \ underline {9} = 9

It results from it that:

9x = 9 \ Rightarrow X = 1

Explanation

The against-intuitive side of this reasoning is due to the fact that, in our spirit, the writing 0, \ underline {9} = 0,99999… corresponds to a finished succession of 9 (i.e. 0,9999… 9). Thus the multiplication by 10 then the result of the subtraction shocks the spirit and seems false (which would be it besides if the continuation of 9 were finished).

Second demonstration (via fractions)

One poses the equality resulting from the algorithm of the Division:

\ frac {1} {3} = 0, \ underline {3}

While multiplying by 3, it comes:

\ frac {3} {3} = 3 \ times 0, \ underline {3}

It follows that:

1 = 0, \ underline {9}

Third demonstration (with a series)

Formalization of 0,99999…

For a more rigorous demonstration, it is necessary to start by defining perfectly what is 0,999…

By writing 0,99999… = 0,9 + 0,09 + 0,009 +…, 0,99999 are defined… as a geometrical Série of first term has = 0,9 and of reason Q = 1/10.

As follows: 0, \ underline {9} = 0,99999 \ ldots = \ lim_ {N \ to \ infty} \ sum_ {i=0} ^ {N} 0,9 \ times \ frac {1} {10^i}

Demonstration by the limit of the series

One can easily show that the sum of N first terms of a geometrical series of reason Q and first term has is worth: S_n = has \ times \ frac {1-q^n} {1-q}

This sum tends towards a limit for N tending towards the infinite one, if and only if Q is strictly smaller than 1, and this limit is then: S = \ frac {has} {1-q}

Here, has = 0,9, Q = 1/10, Q is smaller than 1, therefore the limit exists and is worth S = \ frac {0,9} {1-1/10} = 1

The paradox illustrated by the example of the unit is that all decimal Number, i.e. admitting a finished decimal development, also admits an infinite development (formed only of 9 starting from a certain row). The finished development is the clean writing, that comprising an infinity of 9 is the unsuitable writing. Finally, in fact the objects apparently simplest in decimal writing offer worst complexities: it is believed that 1 is simpler to write in decimal writing than pi, and yet pi admits a single writing, whereas 1 admits two of them. (!)

It is important to remember that the decimal writing is only one in the multiple manners of representing a number in mathematics.

See too

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