I think I disproved mathematics?

Question

Ok, I came across this in maths the other day. Say
X=.9'(recuring) So then
10X=9.9' so then subract the first equation from the second,
10X-X=9X and 9.9' -.9' = 9 meaning
9X=9. Now divide that by 9, and
X=9/9
X=1. Makes sense right? However, when we started this we new that X=.9', so if X= both 1 and .9' , that means that 1=.9'
1 is therefore not constant, so that means that maths does not work. Can someone help me with this?

Answer 1

That's because 1 *is* equal to 0.9(repeating).

Real numbers have two representations.

A lot of people won't grasp the concept that 1 = 0.9(repeating), but this wikipedia article will prove it: http://en.wikipedia.org/wiki/0.999_%3D_1

The equality has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of mathematics education have studied the reception of this equation among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number, an expectation that infinitesimal quantities should exist, that arithmetic may be broken, an inability to understand limits or simply the belief that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There is even a system in which an object that can reasonably be called "0.999…" is strictly less than 1.

Answer 2

This does not prove that math doesn't work. It happens to be a fact that 0.9999... (normally printed as .9 with a bar over the 9, meaning that it's an infinitely repeating decimal) is EQUAL to 1. They are two different ways of writing the same value, just as 1/2 + 1/2, or 6 minus 5, are other ways of writing the save value.

1 is certainly constant. 0.9999.... is the same constant in a different form.

It's troubling for some people to accept, but it's mathematical fact. And that's not just because I say so. See the link below. And if you're still skeptical, consult a mathematician at a local university. I guarantee they will agree that 0.9999.... = 1.

Answer 3

Actually, .999 repeatedly is in fact equal to exactly one. We all know that 1/3 is equal to .333 repeatedly, right? Set up this equation.

1/3=.333 (repeatedly)
Multiply both sides by 3
1=.999 (repeatedly)

Next proof:
A repeating decimal is something over 9, or 99, or 999, etc. For example, 1/9= .111111..., 2/9=.222222..., 1337/9999=.13371337..., and the like. So to get .9 (repeatedly), you put 9/9. 9/9 is obviously the same as one.

Proof three involves limits. I'm not going to go into it, but if you want, look at it here:

http://en.wikipedia.org/wiki/0.999...

And this site has some good info.

http://sprott.physics.wisc.edu/Pickover/pc/9999.html

Answer 4

You aren't making any sense. Evolution is completely random. It appears to happen in spurts and especially when a population becomes very small from whatever reasons the environment throws out. And one other thing....... there IS A WAY to observe the process of evolution in our DNA - its called the Mitochondrial DNA (mtDNA) lineage. It has been used to trace dogs all the way back to prehistoric wolves. In humans, scientists have isolated the Mitochondrial Eve - the single woman that spawned the entire lineage of modern humans (h.sapien.sapiens) around 200,000 years ago. And thats not all..... she and her decendants continued to mate with the previous ancestral group for roughly 50,000 years. In other words, Eve came first, followed by Adam 50,000 years later.

Answer 5

No, you didn't disprove math. You proved that 0.999 . . . = 1. That is a true statement. Are you sure you didn't copy that proof from somewhere? Not that it's so hard that you couldn't figure it out, but it's odd that you'd just randomly think of such a thing. Good night!

Hmmm . . . I don't see what I said that deserved a thumbs down. I only gave good observations and I didn't insult anyone.

Answer 6

I "discovered" this too. At first, I thought it was proof (by contradiction) that 0.9999... is not a rational number, the only repeating decimal that is not a rational number. But according to the wikipedia article below, mathematicians say that it is equal to 1, as you and I "discovered."

Answer 7

You're forgetting something. Whenever you have an infinite repeating number, as in .99 repeating, it's ALWAYS rounded up. .99 repeating is NOT the same number as just .9, and that's what you need to remember. In ALL mathematics where you need to solve for a variable, if you have an infinite repeating number, you MUST round up or down to figure out your answer. So no, to answer your question, you didn't disprove anything.

Answer 8

My dad pulled a similar "trick" on me decades ago that "proved" that 1 + 1 did NOT equal 2, but after closer review, one of the arithmetical steps in the "proving" process was not mathematically allowable, so, you didn't "disprove" anything. You're just trying to pull a trick that I'm too lazy to specify since I've seen it done decades ago. God Bless you.

Answer 9

two numbers are diff when there are other numbers between them. in this case it can be proved that there exist no number between .9' and 1 so they are basically same number.

Answer 10

here's the dilemma:

try subtracting: 1-0.9'(recurring)

what's the difference?how significant?