Does .999… = 1? The inquiry invites the curiosity that students and also the ire of pedants. A well known joke illustrates mine point:

A guy is lost at sea in a hot air balloon. That sees a lighthouse draw close in the fog. “Where am I?” the shouts desperately with the wind. “You’re in a balloon!” he hears as he drifts off right into the distance.

You are watching: Is .9 repeating equal to 1

The response is correct yet unhelpful. When world ask about 0.999… castle aren’t saying “Hey, might you find the border of a convergent series under the axioms the the actual number system?” (Really? Yes, Really!)

No, yes sir a broader, more interesting subtext: What happens as soon as one number it s okay infinitely close come another?

It’s a rare thing when civilization wonder around math: let’s usage the opportunity! instead of bluntly providing technical definitions to meet some require for rigor, let’s permit ourselves to explore the question.

Here’s my fast summary:

The meaning of 0.999… relies on our assumptions around how numbers behave.A typical assumption is that numbers can not be “infinitely close” with each other — they’re one of two people the same, or they’re not. Through these rules, 0.999… = 1 since we don’t have a method to represent the difference.If we permit the idea of “infinitely nearby numbers”, then yes, 0.999… can be less than 1.

Math have the right to be around questioning assumptions, pushing boundaries, and also wondering “What if?”. Stop dive in.

Do Infinitely small Numbers Exist?

The definition of 0.999… is a tricky concept, and depends on what we enable a number to be. Here’s an example: go “3 – 4” typical anything come you?

Sure, it’s -1. Duh. However the inquiry is only straightforward because you’ve embraced the advanced idea of negatives: she ok v numbers gift less than nothing. In the 1700s, once negatives were brand new, the concept of “3-4” was eyed with good suspicion, if allowed at all. (Geniuses of the time thought negatives “wrapped around” after girlfriend passed infinity.)

Infinitely little numbers confront a similar predicament today: they’re new, an obstacle some long-held assumptions, and also are considered “non-standard”.

So, perform Infinitesimals Exist?

Well, do an unfavorable numbers exist? negative exist if you enable them and also have continual rules for your use.

Our present number device assumes the long-standing Archimedean property: if a number is smaller sized than every various other number, it need to be zero. Much more simply, infinitely small numbers don’t exist.

The idea have to make sense: numbers should be zero or not-zero, right? Well, that “true” in the same way numbers have to be over there (positive) or not there (zero) — it’s true since we’ve implicitly excluded various other possibilities.

But, that no issue — let’s see where the Archimedean residential property takes us.

The traditional Approach: 0.999… = 1

If we assume infinitely small numbers don’t exist, us can present 0.999… = 1.

First off, we require to figure out what 0.999… means. Most mathematicians see the problem like this:

0.999… represents a series of numbers: 0.9, 0.99, 0.999, 0.9999, and so onThe question: does this series get so close (converge) come a result that us cannot tell the apart?

This is the reasoning behind limits: Does our “thing come examine” gain so suturing close to another number that we can’t tell them apart, no matter how hard we try?

“Well,” girlfriend say, “How carry out you tell numbers apart?”. Great question. The simplest way to compare is come subtract:

if a – b = 0, lock the sameif a – b is no zero, castle different

The idea behind limits is to find some suggest at i beg your pardon “a – b” i do not care zero (less than any number); the is, we can’t call the “number come test” and also our “result” together different.

The Error Tolerance

It’s still difficult to compare items when they take such different forms (like an limitless series). The following clever idea behind limits: define an error tolerance:

You give me your tolerance for error / accuracy level (call it “e”)I’ll view whether i can obtain the two points to loss within that toleranceIf so, they’re equal! If us can’t tell castle apart, no issue how difficult we try, they have to be the same.

Suppose I market you a raisin granola bar, claiming that 100 grams. You take it it home, examine the non FDA-approved wrapper, and decide to view if i’m lying. You put the snack on her scale and it reflects 100 grams. The range is exact to 1 gram. Did ns trick you?

You can not know: as far as you can tell, within her accuracy, the granola bar is indeed 100 grams. Our present problem is similar: I’m selling you a “granola bar” weighing 1 gram, however sneaky me, i’m actually giving you one weighing 0.999… grams. Deserve to you call the difference?

Ok, let’s occupational this out. Expect your error tolerance is 0.1 gram. Then if you ask for 1, and I give you 0.99, the distinction is 0.01 (one hundredth) and you don’t recognize you’ve been tricked! 1 and also .99 look the very same to you.

But that’s child’s-play. Let’s say your scale is accurate to 1e-9 (.000000001, a billionth that a gram). Fine then, I’ll offer you a candy bar that is .999999999999 (only one trillionth the a gram off) and you’ll be fooled again! Hah!

In fact, rather of choose a certain tolerance choose 0.01, let’s usage a basic one (e):

Error tolerance: eDifference: Well, expect e has “n” number of precision. Permit 0.999… increase until we have actually a difference requiring n+1 number of precision to detect.Therefore, the yongin can always be less than e! and also the difference shows up to it is in zero.

See the trick? this is a visual means to represent it:

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The right line is what she expecting: 1.0, the perfect granola bar. The curve is the number of digits we increase 0.999… to. The idea is to broaden 0.999… until it falls within “e”, your tolerance:

*

At part point, no matter what you pick for e, 0.999… will obtain close sufficient to meet us mathematically.

(As an aside, 0.999… isn’t a growing process, that a final an outcome on that own. The curve to represent the idea that we deserve to approximate 0.999… with better and far better accuracy — this is fodder for another post).

With limits, if the difference in between two points is smaller sized than any type of margin we can dream of, they should be the same.

Assuming Infinitesimals Exist

This an initial conclusion might not sit well with you — you might feel tricked. And also that’s ok! we seem to be ignoring something vital when us say that 0.999… equals 1 because we, with our limited precision, cannot tell the difference.

Newer number systems have arisen the idea that infinitesimals exist. Specifically:

Infinitely small numbers can exist: they no zero, yet look like zero come us.

This seems to be a confound idea, however I view it favor this: atoms don’t exist to cavemen. Once they’ve cut a rock into grains of sand, they can go no further: that’s the the smallest unit they can imagine. Things are either grains, or not there. They can not imagine the concept of atom too small for the naked eye.

Compared to other number systems, we’re cavemen. What we call “tiny numbers” are actually gigantic. In fact, there deserve to be an additional “dimension” of number too little for us to finding — numbers that differ only in this tiny dimension look similar to us, but are various under one infinitely powerful microscope.

I analyze 0.999… favor this: can we do a number a bit less than 1 in this new, infinitely tiny dimension?

Hyperreal Numbers

Hyperreal numbers are one device that provides this “tiny dimension” to study infinitely little numbers. In this, infinitesimals space usually dubbed “h”, and also are thought about to be 1/H (where huge H is infinity).

So, the idea is this:

0.999… 0.999… + h = 1 0.999… = 1 – h

So, 0.999… is simply a tiny bit less than 1, and the distinction is h!

Back come Our Numbers

The trouble is, “h” doesn’t exist back in our macroscopic world. Or rather, h watch the exact same as zero to us — we can’t tell that it’s a tiny atom, not the absence of any type of matter altogether. Here’s one way to visualize it:

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When us switch ago to ours world, it’s dubbed taking the “standard part” that a number. It essentially way we throw away all the h’s, and convert them come zeroes. So,

0.999… = 1 – h St(0.999…) = St(1 – h) = St(1) – St(h) = 1 – 0 = 1

The happy damage is this: in a an ext accurate dimension, 0.999… and also 1 room different. But, when we, through our finite accuracy, shot to define the difference, we cannot: 0.999… and 1 watch identical.

Lessons Learned

Let’s hop back to our world. The objective of “Does 0.999… equal 1?” is not come spit back the answer to a limit question. That’s interpreting the query as “Hey, within our system what walk 0.999… represent?”

The inquiry is about exploration. The really, “Hey, i’m wondering around numbers infinitely close with each other (.999… and 1). How do we manage them?”

Here’s my response:

Our idea that a number has evolved over countless years come include new concepts (integers, decimals, rationals, reals, negatives, imaginary numbers…).In our current system, we haven’t allowed infinitely little numbers. Together a result, 0.999… = 1 due to the fact that we don’t allow there to be a gap between them (so they have to be the same).In various other number solution (like the hyperreal numbers), 0.999… is much less than 1. Here, infinitely tiny numbers are allowed to exist, and this tiny difference (h) is what off 0.999… from 1.

There are life class here: have the right to we prolong our psychological model the the world? Negatives provided us the conception the every number have the right to have an opposite. And you recognize what? It transforms out matter can have an opposite also (matter and also antimatter annihilate every other as soon as they come in contact, similar to 3 + (-3) = 0).

Let’s think about infinitesimals, a tiny dimension beyond our accuracy:

Some theories of physics referral tiny “curled up” size which are embedded into our own. This dimensions might be infinitely little compared to our very own — us never notification them. To me, “infinitely tiny dimensions” space a method to explain something i beg your pardon is there, yet undetectable to us.The physical scientific researches use “significant figures” and also error margins come specify the natural inaccuracy of our calculations. We know that truth is different from what we actually measure: infinitesimals aid make this distinction explicit.

Math no just around solving equations. Expanding our perspective v strange new ideas helps disparate topics click. Nothing be fear wonder “What if?”.

Appendix: where the Rigor?

When writing, I favor to envision a super-pedant, concerned more with satisfying (and demonstrating) his rigor than educating the reader. This mythical(?) nemesis inspires me to emphasis on intuition. Ns really should offer Mr. Rigor a name.

See more: Where Does Oscar Dela Hoya Live ? Oscar De La Hoya

But, rigor has a use: it help ink the pencil-lines we’ve sketched out. I’m no a mathematician, but others have actually written about the details that interpreting 0.999… and also 1 or less than 1:

“So lengthy as the number system has not been specified, the students’ hunch the .999… can loss infinitesimally brief of 1, have the right to be justification in a mathematically rigorous fashion.”

My score is come educate, entertain, and spread attention in math. Deserve to you think that a much more salient method to gain non-math majors interested in the concepts behind analysis? boundaries aren’t going to sector themselves.