# Why does the EUR/USD exchange rate is in fact USD/EUR from a mathematical point of view? Why finance does not use the mathematical notation?

Why does the EUR/USD exchange rate is in fact USD/EUR from a mathematical point of view?

By not using the mathematical notation it's a nightmare to compare exchange rates, so why is everyone still doing it?

Edit: I think I wasn't clear enough, what I would like to understand is why or how the dominant notation came to be the conventional way, and who initiated it. If people agree it's confusing, then why not change it?

• I agree with you, it is confusing at first. It is best to write EURUSD (with no slash between them) and to understand this as a price quotation for the commodity 'EUR' (the first triplet) expressed in or paid by means of 'USD' (the second triplet). This is the language that FX traders use. In the International Economics literature it is written $\frac{USD}{EUR}$ (which you can read as 'dollars per euro'). You just have to be familiar with both notations and be able to switch between them. Jun 25, 2018 at 17:41
• That / symbol doesn’t represent division here, any more than it does in, say dd/mm/yyyy date notation. To remember the convention for how to interpret an FX quote, I find it helpful to thing of an amount of money “moving” from the left to the right via multiplication. E.g we “move” 500 EUR into the USD equivalent amount by multiplying 500 by the EURUSD rate. I still have to think about it though, and I’ve been working with this stuff for some 13 years now. Jun 27, 2018 at 21:16
• It is ok to have a consistent quote convention that is not mathematical: $/$ is not a division. Imagine young mathematicians running around on a trading floor trying to tell FX traders that they have been inconsistent in all those decades. Jan 26 at 11:12
• @JanStuller . Let's better not change those conventions. Trying to do so would lead to chaos because there will surely be someone offering $130$ mn EUR at one million Japanese Yen. Needless to say that this goes to the users who found it illogical. Jan 26 at 11:32
• @JanStuller if you ask people who don't know anything about FX rate and they read EUR/USD, I bet you a majority of ppl will say it is the number of EUR per USD. As well, if you do a dimension analysis as in physics, "stock in USD x FX(EUR/USD)", it would naturally simplify into a stock denominated in EUR . Have said that, the convention is never going to change Jan 26 at 12:35

(1) Suppose 1 EUR = 1.08 USD. Here, EUR and USD are units of measure in que value dimention.

(2) Dividing by USD, EUR/USD = 1.08, which is the notation used by traders. Intuitively, 1 EUR is worth more than 1 USD by a factor of 1.08. In other words, the value of 1 EUR is 1.08 times the value of 1 USD.

Example of a computation using this notation. Let the exchange rate of the euro in terms of the dollar be EUR/USD = 1.08 and let the exchange rate of the dollar in terms of Mexican pesos be USD/MXN = 17.22. Multiplying the corresponding sides of the previous two equations, we get the currency rate of the euro in terms of the Mexican Peso: (EUR/USD)(USD/MXN) = 1.08 × 17.22, or EUR/MXN = 18.5976.

(3) Dividing (1) by EUR, 1 = 1.08 USD/EUR.

The right hand of the last equation is called a conversion factor by physicists. Because any conversion factor equals 1, it can multiply any quantity measuring value without altering its value. For example,

1000 EUR = 1000 EUR × 1 = 1000 EUR × 1.08 USD/EUR = 1080 USD ,

where the second equality follows from (3).

Conclusion (following the second comment by the OP below): Most of us are accustomed to interpreting EUR/USD as units of measurement accompanying a number, as in (3). So, if someone says that EUR/USD is 1.08, the first instinct of someone who is not a currency trader is to interpret that statement as 1.08 EUR/USD, which is wrong because the correct ratio of units that accompanies 1.08 is USD/EUR, hence 1.08 USD/EUR. However, if we read that EUR/USD is 1.08, we should instead interpret it as EUR/USD = 1.08, or the division of the value of 1 EUR by that of 1 USD.

Note: The argument above is the same used for physical quantities. The following is an example in length dimension. Using the same numbering as above,

(1) Let 1 m = 100 cm

(2) Dividing by cm, m/cm = 100. A meter is 100 times the length of a centimeter.

(3) Dividing (1) by 100 cm, 1 m / (100 cm) = 1

Reexpressing in meters a quantity expressed in centimeters:

76 cm = 76 cm × 1 = 76 cm × 1 m / (100 cm) = 0.76 m ,

where the second equality follows from (3).

• Your arguments are very interesting. But I am not sure your mathematics is standard, specifically the way you are interpreting units of measurement as if they were variables. Nevertheless, regardless of how mathematically correct it is, you have given me a satisfying explanation of how that notation may have originated from mathematics. Thank you Jan 26 at 22:04
• I think your math is correct, this very enlightening. I was trapped into interpreting EUR/USD as units of measurement accompanying a number, basically if someone said to me that EUR/USD was 1.08, my intuition was to interpret that statement as 1.08 EUR/USD which obviously seemed wrong to me, because the correct units that accompany that 1.08 is in fact USD/EUR, hence 1.08 USD/EUR. But what you are telling me is that if I read somewhere that EUR/USD is 1.08, I should instead interpret it as EUR/USD = 1.08, basically as if dividing the inherent value of 1 EUR by 1 USD. Thank you again ^_^ Jan 26 at 22:41

I can give an answer the second part of your edit. In my experience terminology, once established and active with a critical mass tends to propagate and then remain. It doesn't really change much. Particularly in trading, terminology has evolved around the central points of liquidity, and becomes embedded in many systems; computerised, vocal and written contracts. Often it forms based on the fastest way to express something when agreeing a trade. In the cross-currency swap market for example I speculate that the sole reason that the voice broker market quotes "euro-dollar" (in that order of currencies) as opposed to "dollar-yen", when they are technically the same similar product, is simply the nature that it rolls off a western tongue faster and easier than the converse. And in the dollar-yen case it avoids having to have notionals like "1.1 trillion yen please", cause you default to USD.

As a concrete example, some terminology that appeared in the EUR IRS interdealer market 2y ago is the name of the price "10Y gadget", which is the yield differential between 10Y bond futures and 10Y swaps. I found this an awful name and lobbied to change it to "10Y spread", which is common in GBP and USD. But when you find there is no real (simultaneous) will to change it doesn't get much traction and as long as everyone uses it and understands it you just have to go with it.

Perhaps there is no legitimate explanation for the appearance of the '/', and after battling the initial confusion no one cares anymore.

You ask how the notation EUR/USD = 1.2 originated. To put it more bluntly how could anyone have thought up to write the exchange rate in such an illogical fashion, which seems backwards.

You can think of it not as an exchange rate between currencies but as a comparison of the "strength" of one currency to the other. In the old days 'strength' might have been based on a comparison of the amount of gold backing one unit of each currency (so eur/usd = 1.2 says that a euro has 20% more ounces of gold in it than a dollar). Or, when gold ceased to be used, 'strength' might be based on a basket of goods which a unit of currency can purchase (Gustav Cassel's Purchasing Power theory). So EUR/USD = 1.2 says with one EUR you can buy 1.2 times the consumer goods basket than you can buy with 1 USD. Or 1.2 times as many barrels of oil, if you prefer.

Then it makes sense. And dm63 made a similar point here in a comment. But I am not sure if this will help or confuse things further.

• Interesting! Although "strength" in this case is rather numerical than real, because the price level isn't directly linked to the currency exchange rate. In other words, it's not guaranteed that for an amount in one currency in one country you can buy the same things as you can buy in the other country after converting the amount to the other currency. For example, being able to buy bread for 1 EUR in Europe doesn't guarantee it's 1.2 USD in USA. Even within Europe the bread might cost more or less in different countries having the same currency. So, "strength" here isn't that meaningful.
– mdcq
Apr 9 at 12:49