In Bitcoin

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Myths
Let's clear up some common Bitcoin misconceptions.
Bitcoin is just like all other digital currencies; nothing new
Nearly all other digital currencies are centrally controlled. This means that:

They can be printed at the subjective whims of the controllers
They can be destroyed by attacking the central point of control
Arbitrary rules can be imposed upon their users by the controllers
Being decentralized, Bitcoin solves all of these problems.

Bitcoins don't solve any problems that fiat currency and/or gold doesn't solve
Unlike gold, bitcoins are:

Easy to transfer
Easy to secure
Easy to verify
Easy to granulate
Unlike fiat currencies, bitcoins are:

Predictable and limited in supply
Not controlled by a central authority (such as The United States Federal Reserve)
Not debt-based
Unlike electronic fiat currency systems, bitcoins are:

Potentially anonymous
Freeze-proof
Faster to transfer
Cheaper to transfer
Miners, developers or some other entity could change Bitcoin's properties to benefit themselves
Bitcoin's properties cannot be illegitimately changed as long as most of bitcoin's economy uses full node wallets. Transactions are irreversible and uncensorable as long as no single coalition of miners has more than 50% hash power and the transactions have an appropriate number of confirmations.

Bitcoin requires certain properties to be enforced for it to be a good form of money, for example:

Nobody ever created money out of nothing (except for miners, and only according to a well-defined schedule).
Nobody ever spent coins without knowing their private key.
Nobody spent the same coin twice
Nobody violated any of the other tricky rules that are needed to make the system work (difficulty, proof of work, DoS protection, ...).
These rules define bitcoin. A full node is software that verifies the rules of bitcoin. Any transaction which breaks these rules is not a valid bitcoin transaction and would be rejected in the same way that a careful goldsmith rejects fool's gold.

Full node wallets should be used by any intermediate bitcoin user or above and especially bitcoin businesses. Therefore anybody attempting to create bitcoins with invalid properties will find themselves being rejected by any trading partners. Note that lightweight wallets and web wallets do not have the low-trust benefits of full node wallets. Lightweight (SPV) wallets will blindly trust the miners, meaning if 51% of miners printed infinite coins or spent the same coin twice then lightweight wallet users would happily accept these fake bitcoins as payment. Web wallets blindly trust the web server which could display anything at all.

Miners are required to choose between multiple valid transaction histories. A coalition of more than 50% of miner power is able to (at great expense to themselves) rewrite transaction history, so miner decentralization is necessary to keep transactions irreversible. Miners burn a lot of electrical power in the mining process so they must constantly be trading their bitcoin income in order to pay bills. This makes miners utterly dependent on the bitcoin economy at large and therefore gives them a strong incentive to mine valid bitcoin blocks that full nodes will accept as payment.

Influential figures in the community (such as developers, politicians or investors) may try to use their influence to convince people to download and run modified full node software which changes bitcoin's properties in illegitimate ways. This is unlikely to succeed as long as counterarguments can freely spread through the media, internet forums and chatrooms. Many bitcoin users do not follow the bitcoin forums on a regular basis or even speak English. All appeals to run alternative software should be looked at critically for whether the individual agrees with the changes being proposed. Full node software should always be open source so any programmer can examine the changes for themselves. Because of the co-ordination problem, there is usually a strong incentive to stick with the status quo.

See also: Full_node#Economic_strength See also this blog post: Who Controls Bitcoin?

Bitcoin is backed by processing power
It is not correct to say that Bitcoin is "backed by" processing power. A currency being "backed" means that it is pegged to something else via a central party at a certain exchange rate yet you cannot exchange bitcoins for the computing power that was used to create them. Bitcoin is in this sense not backed by anything. It is a currency in its own right. Just as gold is not backed by anything, the same applies to Bitcoin.

The Bitcoin currency is created via processing power, and the integrity of the block chain is protected by the existence of a network of powerful computing nodes from certain attacks.

Bitcoins are worthless because they aren't backed by anything
One could argue that gold isn't backed by anything either. Bitcoins have properties resulting from the system's design that allows them to be subjectively valued by individuals. This valuation is demonstrated when individuals freely exchange for or with bitcoins. Please refer to the Subjective Theory of Value.

See also: the "Bitcoin is backed by processing power" myth.

The value of bitcoins are based on how much electricity and computing power it takes to mine them
This statement is an attempt to apply to Bitcoin the labor theory of value, which is generally accepted as false. Just because something takes X resources to create does not mean that the resulting product will be worth X. It can be worth more, or less, depending on the utility thereof to its users.

In fact the causality is the reverse of that (this applies to the labor theory of value in general). The cost to mine bitcoins is based on how much they are worth. If bitcoins go up in value, more people will mine (because mining is profitable), thus difficulty will go up, thus the cost of mining will go up. The inverse happens if bitcoins go down in value. These effects balance out to cause mining to always cost an amount proportional to the value of bitcoins it produces.

Bitcoin has no intrinsic value (unlike some other things)
This is simply not true. Each bitcoin gives the holder the ability to embed a large number of short in-transaction messages in a globally distributed and timestamped permanent data store, namely the bitcoin blockchain. There is no other similar datastore which is so widely distributed. There is a tradeoff between the exact number of messages and how quickly they can be embedded. But as of December 2013, it's fair to say that one bitcoin allows around 1000 such messages to be embedded, each within about 10 minutes of being sent, since a fee of 0.001 BTC is enough to get transactions confirmed quickly. This message embedding certainly has intrinsic value since it can be used to prove ownership of a document at a certain time, by including a one-way hash of that document in a transaction. Considering that electronic notarization services charge something like $10/document, this would give an intrinsic value of around $10,000 per bitcoin.

While some other tangible commodities do have intrinsic value, that value is generally much less than its trading price. Consider for example that gold, if it were not used as an inflation-proof store of value, but rather only for its industrial uses, would certainly not be worth what it is today, since the industrial requirements for gold are far smaller than the available supply thereof.

In any event, while historically intrinsic value, as well as other attributes like divisibility, fungibility, scarcity, durability, helped establish certain commodities as mediums of exchange, it is certainly not a prerequisite. While bitcoins are accused of lacking 'intrinsic value' in this sense, they make up for it in spades by possessing the other qualities necessary to make it a good medium of exchange, equal to or better than commodity money.

Another way to think about this is to consider the value of bitcoin the global network, rather than each bitcoin in isolation. The value of an individual telephone is derived from the network it is connected to. If there was no phone network, a telephone would be useless. Similarly the value of an individual bitcoin derives from the global network of bitcoin-enabled merchants, exchanges, wallets, etc... Just like a phone is necessary to transmit vocal information through the network, a bitcoin is necessary to transmit economic information through the network.

Value is ultimately determined by what people are willing to trade for - by supply and demand.

Bitcoin is illegal because it's not legal tender
In March 2013, the U.S. Financial Crimes Enforcement Network issues a new set of guidelines on "de-centralized virtual currency", clearly targeting Bitcoin. Under the new guidelines, "a user of virtual currency is not a Money Services Businesses (MSB) under FinCEN's regulations and therefore is not subject to MSB registration, reporting, and record keeping regulations." Miners, when mining bitcoins for their own personal use, aren't required to register as a MSB or Money Transmitter.

In general, there are a number of currencies in existence that are not official government-backed currencies. A currency is, after all, nothing more than a convenient unit of account. While national laws may vary from country to country, and you should certainly check the laws of your jurisdiction, in general trading in any commodity, including digital currency like Bitcoin, BerkShares, game currencies like WoW gold, or Linden dollars, is not illegal.

Bitcoin is a form of domestic terrorism because it only harms the economic stability of the USA and its currency
According to the definition of terrorism in the United States, you need to do violent activities to be considered a terrorist for legal purposes. Recent off-the-cuff remarks by politicians have no basis in law or fact.

Also, Bitcoin isn't domestic to the US or any other country. It's a worldwide community, as can be seen in this map of Bitcoin nodes.

Bitcoin will only enable tax evaders which will lead to the eventual downfall of civilization
Cash transactions offer an increased level of anonymity, yet are still taxed successfully. It is up to you to follow the applicable tax laws in your home country, or face the consequences.

While it may be easy to transfer bitcoins pseudonymously, spending them on tangibles is just as hard as spending any other kind of money anonymously. Tax evaders are often caught because their lifestyle and assets are inconsistent with their reported income, and not necessarily because government is able to follow their money.

Finally, the Bitcoin block chain is a permanent record of all transactions, meaning it can be mined for info at any time in the future making investigation, tracing of funds, etc much easier than with other forms of payment.

Bitcoins can be printed/minted by anyone and are therefore worthless
Bitcoins are not printed/minted. Instead, blocks are computed by miners and for their efforts they are awarded a specific amount of bitcoins and transaction fees paid by others. See Mining for more information on how this process works.

Bitcoins are worthless because they're based on unproven cryptography
SHA-256 and ECDSA which are used in Bitcoin are well-known industry standard algorithms. SHA-256 is endorsed and used by the US Government and is standardized (FIPS180-3 Secure Hash Standard). If you believe that these algorithms are untrustworthy then you should not trust Bitcoin, credit card transactions or any type of electronic bank transfer. Bitcoin has a sound basis in well understood cryptography.

Early adopters are unfairly rewarded
Early adopters are rewarded for taking the higher risk with their time and money. The capital invested in bitcoin at each stage of its life invigorated the community and helped the currency to reach subsequent milestones. Arguing that early adopters do not deserve to profit from this is akin to saying that early investors in a company, or people who buy stock at a company IPO (Initial Public Offering), are unfairly rewarded.

This argument also depends on bitcoin early adopters using bitcoins to store rather than transfer value. The daily trade on the exchanges (as of Jan 2012) indicates that smaller transactions are becoming the norm, indicating trade rather than investment. In more pragmatic terms, "fairness" is an arbitrary concept that is improbable to be agreed upon by a large population. Establishing "fairness" is no goal of Bitcoin, as this would be impossible.

Looking forwards, considering the amount of publicity bitcoin received as of April 2013, there can be no reasonable grounds for complaint for people who did not invest at that time, and then see the value (possibly) rising drastically higher.

By starting to mine or acquire bitcoins today, you too can become an early adopter.

21 million coins isn't enough; doesn't scale
One Bitcoin is divisible down to eight decimal places. There are really 2,099,999,997,690,000 (just over 2 quadrillion) maximum possible atomic units in the bitcoin system.

The value of "1 BTC" represents 100,000,000 of these. In other words, each bitcoin is divisible by up to 108.

As the value of the unit of 1 BTC grew too large to be useful for day to day transactions, people started dealing in smaller units, such as milli-bitcoins (mBTC) or micro-bitcoins (μBTC).

Bitcoins are stored in wallet files, just copy the wallet file to get more coins!
No, your wallet contains your secret keys, giving you the rights to spend your bitcoins. Think of it like having bank details stored in a file. If you give your bank details (or bitcoin wallet) to someone else, that doesn't double the amount of money in your account. You can spend your money or they can spend your money, but not both.

Lost coins can't be replaced and this is bad
Bitcoins are divisible to 0.00000001, so there being fewer bitcoins remaining is not a problem for the currency itself. If you lose your coins, indirectly all other coins are worth more due to the reduced supply. Consider it a donation to all other bitcoin users.

A related question is: Why don't we have a mechanism to replace lost coins? The answer is that it is impossible to distinguish between a 'lost' coin and one that is simply sitting unused in someone's wallet. And for amounts that are provably destroyed or lost, there is no census that this is a bad thing and something that should be re-circulated.