Without mathematical formulas, 0.999.... Does NOT Equal 1

1. Things that look different are not necessarily unequal To show that two things can look different and yet be equal, I chose to use dogs in order to minimize any use of numbers to avoid any accusations of a mathematical argument. However, I acknowledge your point that they are two separate breeds of dog, and therefore not completely equal. Therefore, I will use another analogy to show that two things CAN be equal while looking completely different. This is obvious to anyone that has any familiarity with Roman numerals. Roman numerals are represented by letters of the Phoenician alphabet, and look completely different from the Arabic numerals that we use today. If one were to write any number larger than one in Roman numerals and then in Arabic numerals, the result will be two completely equal numbers that look different from one another. I believe this example shows that the argument of 0.999.... and 1 looking different shows nothing, as we can find many equal values that look different and yet remain equal. The KCA No person can ever write out 0.999.... and no computer can ever display this number. On this point, I wholeheartedly agree with you. However, I disagree with your conclusion. The fact that no person, machine, or anything else could ever possibly fully write out 0.999... does not mean that infinites do not exist. In fact, the very point you make actually shows that infinites do exist. If a person, machine, or anything else were ever able to write out all of 0.999... then, no matter how large it is, it would be finite by definition. The fact that this feat cannot be accomplished shows that infinites do exist. In addition, I have used a real life example to show the existence of infinites. The universe is infinite! No person, machine, or anything else could ever possibly explore ALL of the universe. It is impossible. But just because this feat is impossible, it does not mean that the universe does not exist. It is obvious that the universe does exist. The universe is also an infinite. Therefore, infinites DO exist. Pharell's Hot Air Balloon Regardless of the song lyrics or rainbows, the hot air balloon does not make it to space. Pharell is most likely on LSD if he thinks a magical rainbow will support his weight (a rainbow is a simple dispersion phenomenon and not a tangible object). But even if his hot air balloon did make it into space, a 0.999.... hot air balloon in such a case would also make it to space. They are one and the same. The song lyrics do nothing to refute the equality of 1 and 0.999... The Rage Man In this example, the man is shouting "FFFFFFFFFFFFUUUUUUUUUUUUUUUU-". From here, you assign values to the letters (and place a decimal point between the final F and the first U). But the argument that was created from this holds no water. The fact that he is not shouting "1" in this example, but "0.999...." does not show that the two values are unequal. In fact, it seems that you have merely taken your very first argument (i.e. they look different therefore they are different) and converted it from a visual standpoint to an auditory one. In essence, the Rage Man example simply claims that since the two sound different they are not equal. But this argument contains the same flaw as the visual example. If an American screams out "Eighty eight!" and a Japanese man screams out "Hachi ju hachi!" the two sounds produced will be completely different. Yet the value of what is screamed by each remains the same. So the fact that this man screaming "0.999...." in code is not screaming "1" instead does not show that the two are unequal. Just as two values can look different yet remain equal, two values can also sound different and remain equal. The remaining arguments of 1111111.9999999.... and 00000.9999999.... not being equal is obvious. These are not the values we were debating as equal (nor is 000.1111...) and so this section I will justifiably ignore. My argument To reiterate, any two things that are unequal must be different in some way. Yet when looking at the difference between 0.999... and 1, we have no difference in value. As pointed out by you, we do have differences in visual appearance and in the way they sound when spoken (or shouted). But the difference in value, if one were asked to show it, is simply 0.00000.... If there is no difference shown between these two numbers, they are of equal value. Therefore, 0.999.... does, in fact, equal 1.