Is 3/3 the same as 1 or not? The Confusion Between Decimal Notation and Numbers

[dracobook]

My friend thought of this some time ago...
1/3 is equal to .333333...
2/3 is equal to 2*(1/3) or .66666...
3/3 is equal to 3*(1/3). If 1/3 is equal to .333... then wouldn't 3/3 be equal to .9999...? Also, isn't .9999... not the same as 1?

 

[amcavoy]

You should search this. Analogues of this question have come up before.

 

[Hurkyl]

Also, isn't .9999... not the same as 1?

Double negatives! I don't think you're asking quite what you meant to ask.

0.999~ is the same number as 1. That the same number can have different symbolic representations shouldn't be surprising... after all, you know that the fractions 1/2 and 3/6 are the same number!

Examples like the one you posted are demonstrations of why mathematicians decided to require that 0.999~ = 1 in the decimal number system: otherwise arithmetic would not work nicely.

 

[HallsofIvy]

Double negatives! I don't think you're asking quite what you meant to ask.

Actually, Hurkyl, reading the whole post I think this time the double negative was exactly what he meant. Many people who confuse the "decimal notation" for numbers with the numbers themselves think that 0.9999... is not 1. Here I think the OP was saying: "Isn't it the case that 0.9999... is not the same as 1" and the double negative is perfectly correct.