0.99999…. = 1. That’s true, apparently. Not convinced? Our intuitions (for most non-mathematicians) is to reject the equality of these two numbers. We feel, somehow, that 0.99999… must be less than 1, and that no matter how small the difference, logic just insists that there just is a difference. We might make various appeals to logic or common-sense before possibly making complex arguments about infinity and the meaning of infinite decimal expansions.
But here’s a proof that the two are equal:
0.33333… = 1/3
1/3 x 3 = 1
0.33333… x 3 = 0.99999…
therefore 1 = 0.99999…
My interest in this simple mathematical oddity is in what it tells us about human psychology. There’s a detailed discussion of it here on wikipedia. We act as intuitive creatures, mostly using approximations to judge angles, amounts, distances, and sizes. We take shortcuts to the truth. Abstract reasoning is relatively difficult and not entirely convincing even when we see it worked out in front of us. Partly because we’re so blind to the unconscious shortcuts our minds take, we don’t even see them as shortcuts. And the other reason is that those shortcuts have been honed by millions of years of evolution to be – actually – quite extraordinarily successful ways of approaching the real world. It would be foolhardy to ditch those modes of reasoning without, er, good reason.
Yet the truth is, sometimes one should ditch the intuitions and trust in reason. You see this most obviously every time a retailer sets prices at $9.99 or sell cars for $39,999. It is as near to $10 or $40,000 as makes no difference and most people know it’s a con. But it tugs at something deep in our subconscious. Why else would the retailer bother doing it? (This is an analogous situation to 1 = 0.99999… only, and is not the same, of course!)
This insight into human obstinacy in the face of logic can be applied to many situations. It has always intrigued me why people gamble – can’t they see that the odds are stacked against them? Well, most gamblers know – at some higher abstract level – that the odds are against them, but a mixture of gut instinct, fear, greed, superstition, and false reasoning about probabilities convince them that they should gamble. I remember well when a syndicate in Ireland became convinced that they could increase their chances of winning the lottery by buying combinations of numbers dictated by extremely erroneous understandings of how lotteries work. They lost thousands of euros in a vain attempt to defy logic.
In an even broader sense, we human beings apply our narrow reactions to the world around us – the common-sense view of the world – to entirely inappropriate situations. We cannot entirely shake the suspicion that things are a particular way because they should or must be a particular way, despite science and logic telling us differently.
For example, many people hold dear to a belief in an afterlife, because they cannot quite shake the internal, subjective sense that they’ve always been around. And they cannot imagine (none of us can) not being around. It’s like looking down at your own funeral – that logically impossible situation is still dreamt by most of us. Logic says it is impossible to have an afterlife. (Or as near impossible as makes no difference…). But then, what’s logic in the face of human need?