Are Decimals Still Relevant in Today's Number System?

[Omega0]

We use the 10 as a base for numbers because we have 10 fingers, right? The 10 commandments, awesome... we have the system of numbers to the power of 2 which makes much more sense to me. Of course feel free to pack them together to octs or hex numbers but wouldn't you say that decimals doesn't make much sense nowadays?

 

[phinds]

... but wouldn't you say that decimals doesn't make much sense nowadays?

No, I absolutely would not say that. Do you seriously think it's easier to write numbers in binary? Do you think Joe the Plumber is ever going to learn octal or hex? Ridiculous, as far as I can see.

 

[fresh_42]

There are several reasons to keep the decimal system:

• Humans can grasp amounts of 1 to 5 without counting, so 5 is a natural unit determined by evolution.
• 1,2,5 are perfect partitions for coins and bills to realize any amount of money with the least number.
• We already tried 36 and 60, but only traces from that are left.
• Binary is too short, so hex would be a natural choice, however, see my previous point.
• Kids still learn to count by using fingers.
• 10 isn't the worldwide base because someone made a decision, it is because it is convenient.
• Did I mention worldwide? It is unlikely that all are simultaneously wrong.
• Even programmers don't talk binary.
• We do use binary: in logic.

 

[Omega0]

Do you seriously think it's easier to write numbers in binary?

It is not easier but it may make much more sense. The base of a number system is just used by convention. Binary has the disadvantage of being a long notation but it is finally a very logical notation.
There is no logic about 10 but the number of our fingers. Even to abbreviate to the base 4 or 8 might make more sense. I personally like 3.

Do you think Joe the Plumber is ever going to learn octal or hex? Ridiculous, as far as I can see.

Sorry but I am astonished that you as a "Science Advisor" call this ridiculous. So far I always had a "relativistic view" on things. Not only about physics but also mathematics - but you, "phinds", seem to have discovered something new about the number 10, amazing.

 

[Omega0]

There are several reasons to keep the decimal system:

• Humans can grasp amounts of 1 to 5 without counting, so 5 is a natural unit determined by evolution.
• 1,2,5 are perfect partitions for coins and bills to realize any amount of money with the least number.

I said that our counting comes from the number of our fingers. The question is if this is useful on the long term - I mean, we are a scientific culture meanwhile.

• We already tried 36 and 60, but only traces from that are left.

I don't get it, what do you mean?

• Binary is too short, so hex would be a natural choice, however, see my previous point.

Sure, binary is short which may make $2^n$ as a base interesting where $n >2$.

• Kids still learn to count by using fingers.

Kids may count the fingers on it's own but we are the ones to give them a base to count.

• 10 isn't the worldwide base because someone made a decision, it is because it is convenient.

Nope, it comes frome the history.

• Did I mention worldwide? It is unlikely that all are simultaneously wrong.

You probably didn't understand: It is not about "wrong". The base for numbering things is interchangeable and having this in mind you should understand that none is "wrong". It is just convention.

• Even programmers don't talk binary.

They do if we are speaking about "tree algorithms" and so on. Binary is pretty much used for digital things.

ree algo

• We do use binary: in logic.

Yes, and nature is pretty logic as far as I can see.

PS: If it so hard to understand what I am talking about I might just state: The system how we count things is not given by sort of God or so. It could be changed. It isn't a constant in the Universe.

I don't know if this would be good but it is possible and it makes sense to at least think about it.

 

[PeroK]

but wouldn't you say that decimals doesn't make much sense nowadays?

What precisely is the problem with base $10$?

 

[phinds]

Sorry but I am astonished that you as a "Science Advisor" call this ridiculous.

Then I absolutely encourage you to use nothing but binary, ocatal, and/or hex from here on out. Good luck with that.

 

[Omega0]

What precisely is the problem with base $10$?

The standard question. "What is the problem with...". What ist the problem with using inches and not meters? It is easy to set $c=1$ but if we are speaking about math then it is a "problem"? 10 is not a problem. I am not speaking about "problems" but about something at least I think about.
There are several problems, in number theory for example as you will know. I believe the question is always how to improve things.
I am not stating that we need to change from base 10 to base 2 or a power of it but perhaps we should think about it. Give your mind some space.
What I like is to think about prime numbers as the base. My favorite is 3.

So, Perok, do what you want, but honestly, I haven't a problem with 10.

 

[symbolipoint]

No, I absolutely would not say that. Do you seriously think it's easier to write numbers in binary? Do you think Joe the Plumber is ever going to learn octal or hex? Ridiculous, as far as I can see.

Base 2 - Simple but not always easy
Base 10 - Complicated but often not as difficult

 

[Omega0]

Base 2 - Simple but not always easy
Base 10 - Complicated but often not as difficult

Base 2: What is "simple"? What is easy about it? It is a mapping of the smallest possible alphabet to the world of numbers. I am pretty much convinced that 95% percent of the human population or so don't find that easy.

Base 10: Why complicated? Sure, not complicated. It is what we are teached. That makes 10 easier. 10 seems to be natural, right?

 

[Borek]

I mean, we are a scientific culture meanwhile.

Say that to "I won't vaccinate myself and the world is run by lizards Joe" (and remember he dominates some populations).

I don't get it, what do you mean?

They were tried, they didn't survive.

If you think about advocating for change try to convince Americans to use metric system.

 

[Frabjous]

We already tried 36 and 60, but only traces from that are left.

Clocks are an obvious example of 60. Do you have any more common 36 or 60 examples?

 

[Bystander]

Clocks are an obvious example of 60. Do you have any more common 36 or 60 examples?

Babylonian recognition of the three-four-five triangle for construction of right angles, sixty.

 

[Frabjous]

Babylonian recognition of the three-four-five triangle for construction of right angles, sixty.

or in binary
Babylonian recognition of the 11-100-101 triangle for construction of right angles, 111100.

 

[256bits]

We use the 10 as a base for numbers because we have 10 fingers, right?

We can count to 12 on one hand, using just the three finger bones on four fingers.

 

[pinball1970]

Nope, it comes frome the history

Not being picky bit there were not 10 commandments there were over 600.
Also 12 is a more prominent and significant number in terms of that history as is 3, 6 and 7
I would also check out the imperial system we had before 1971 in the UK.
4x farthing to one old penny
12 x penny to one shilling
I think there was a two shilling piece a threepenny bit and a sixpence.
I learned this as a kid and somehow made the transition to metric unscathed.
I think they used every number except 10. Same with weights and measures.

 

[PeroK]

I think there was a two shilling piece ..

"i" before "e", except after "c"!

 

[fresh_42]

Clocks are an obvious example of 60. Do you have any more common 36 or 60 examples?

The 360° of a circle.

Maybe the calendar can be added to the list. It would have been more natural to use the moon for the length of a month ...

month (n.) ... Old English monað, from Proto-Germanic *menoth- (source also of Old Saxon manoth, Old Frisian monath, Middle Dutch manet, Dutch maand, Old High German manod, German Monat, Old Norse manaðr, Gothic menoþs "month"), which is related to *menon- "moon" (see moon (n.)). Originally the month was the interval between one new moon and the next (a sense attested from late Old English).

https://www.etymonline.com/search?q=month

... but we use 30, not 28.

Wikipedia has a nice list:
https://en.wikipedia.org/wiki/Sexagesimal#Other_historical_usages

Some usages disappeared quite recently (in historical dimensions).

 

[TeethWhitener]

https://external-preview.redd.it/XI...bp&s=371835f02c3e48d01af3899598decfdb035f2bad

 

[cmb]

We use the 10 as a base for numbers because we have 10 fingers, right? The 10 commandments, awesome...

I think you'll find it was 15 commandments.

There's a video of it here;-

 

[Omega0]

Not being picky bit there were not 10 commandments there were over 600.
Also 12 is a more prominent and significant number in terms of that history as is 3, 6 and 7

I read some sources in the internet and indeed, 12 would be a better choice than 10 (reason: 2 * 3 * 2 = 3 * 4 = 2 * 6). For the same reason 360 and 60 makes sense. Or, with other words: Hex wouldn't be an advantage, at least currently. Or, with other words, my idea wasn't that good.

 

[symbolipoint]

I read some sources in the internet and indeed, 12 would be a better choice than 10 (reason: 2 * 3 * 2 = 3 * 4 = 2 * 6). For the same reason 360 and 60 makes sense. Or, with other words: Hex wouldn't be an advantage, at least currently. Or, with other words, my idea wasn't that good.

Maybe. People would need to put into their long culture. Look: Base Two uses two symbols for integers. This should make for easy computing for powers of two. (Computing , like what you could do in your head or what could be done fairly easily on paper). Base Ten uses ten symbols for integers. This is what we are usually accustomed to. Computing should be easy enough in powers of ten, either in the head or done on paper.

You want Base Twelve to be common? If were done, then surely the culture over time would popularize this, and it would be more common to our thinking, and we'd be taught this from kindergarten through high school, and beyond.

 

[DaveE]

it makes sense to at least think about it.

We did think about it, briefly.

 

[fresh_42]

You want Base Twelve to be common? If were done, then surely the culture over time would popularize this, and it would be more common to our thinking, and we'd be taught this from kindergarten through high school, and beyond.

We can't even agree to drive on the same side of the road! But at least, this is binary.

 

[DaveE]

The number system should be optimized to the hardware processing it. One system will not be optimum for all cases. The human is much more capable of operating with a higher base and fewer digits. While digital electronics is binary at the gate/transistor level, that is often not an efficient way for more complex digital systems to operate. The microprocessor that you are using right now handles those "binary" numbers in large groups, probably 32 at a time. You might think of this as base 4,294,967,296, or base 2; whatever. In a communication channel with limited bandwidth, you may send information serially, 1 bit at a time, you may use morse code, or you may do what your cell phone, Wi-Fi, or satellite TV does and send it in base 4, 8, or 16.

You misunderstood the world we live in from the outset, there isn't a single "best way".

Then there's the whole human culture/psychology/education angle to consider. Why not start first by getting everyone in Switzerland to speak the same language?

 

[cmb]

I trust we are all aware that there are up to 31 days in a month because 31 is as much as you can count on the fingers of two hands (... in binary).



[I might have made that bit up!]

The one amusing detail of using base 10 that I like the most is that we write the numbers the wrong way around.

The 10 digits that came out of Indian/Arabic tradition (ask them where they got that convention from) which we copied in the west were written right to left, per Arabic scripts, smallest digit to largest digit. So one always was meant to know which 'column' that leading digit is in (should be the 'units', smallest, column).

But, hey, we just went on in there, copied the right-to-left Arabic script but in left-to-right text, and ended up with the columns swapped around.

How much easier if we had not misunderstood that bit, for those little kids trying to get the hang of addition if that first column (unit) digit of a number always lined up with the next number!

[I didn't make that bit up!]

 

[Omega0]

The number system should be optimized to the hardware processing it. One system will not be optimum for all cases.

Pretty sure.

The human is much more capable of operating with a higher base and fewer digits.

Sure, it makes sense to not write down only 2 symbols. Around ten may be good but I think the base 12 would be better, easier. In a few hundred years the situation may be completely different. Nowadays it should be clear that only human communication uses base 10. On the lowest OSI levels you can have completely different states. Machines can talk to each other in completely different number systems which makes very much sense because they don't have 10 fingers. (LTE has 60 "fingers" as far as I know)

The microprocessor that you are using right now handles those "binary" numbers in large groups, probably 32 at a time.

Binary is binary. If you have only 2 states you can group them etc. but the numbering is to the base 2.

You misunderstood the world we live in from the outset, there isn't a single "best way".

I misunderstood nothing. I am just head shaking about people who believe that everything is perfect because it is at it is. Things currently are exactly as they should be because they are as they are.

Congratulation!

Then there's the whole human culture/psychology/education angle to consider. Why not start first by getting everyone in Switzerland to speak the same language?

I personally would convince the USA from the metric system which is a running gag in the rest of the world - but I know, it is not easy.
Perhaps even this is a cultural gap between us but as soon as things appear "which are because they are", see the "Luminiferous aether" (even the word is a pain), I personally feel that there might be something going on which needs a change - at least on the long term. Or which could be changed.

I personally accept nothing as a constant which is not a constant.

 

[Frabjous]

I personally would convince the USA from the metric system which is a running gag in the rest of the world - but I know, it is not easy.

Isn’t that a base 10 system?

 

[Omega0]

Isn’t that a base 10 system?

Sure it is base 10 but we spoke about natural languages etc, see above.

 

[jbriggs444]

Before you decide that it would be a good thing to standardize on base 12, be aware that xkcd has something to say about standards.

 

[Omega0]

Before you decide that it would be a good thing to standardize on base 12, be aware that xkcd has something to say about standards.

Good one. PS did you note that 2019 happened a lot?

https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units

I thought this would be a shockwave but it finally felt pretty calm, at least for my audience. No clue how this can't be exciting but the difference might be that you have to learn something, pass exams etc. and me who is over it.

 

[cmb]

The reason why we have the 'Arabic' numerals that we use today is not well explained by any sensible argument.

I mean, there are nine numerals. If we had one numeral per digits of two hands, then we should have ten symbols and base eleven!? The original numbers were just letters, so there wasn't a shortage of those because they had ... however many then had then. (Why do English language users have 26 letters today?)

So we'd be into the realm of speculation why it's not some other number.

Base 10 was found to work well. Napoleon's political successes saw it cemented in European society as a legal thing as he had a personal belief it was good to make that a fixed convention one way or another, which it was.

Of any currently accepted convention that might have a chance of an overhaul before the doom of our civilisation, our use of base 10 would be one of the least likely candidates.

So when we finally make 'first contact' with some extra-terrestrial intelligence, what number base they use will generate some discussion for sure, either 'Whaat!? You use Base 10 too!?" or they ask "Why would you use Base 10 when Base 256 is so much more sensible?" and we discover their kids learn 255 counting numerals at an early age and a multiplication table that would blow our minds! ;)

 

[Omega0]

So when we finally make 'first contact' with some extra-terrestrial intelligence, what number base

I am pretty convinced that it will not be humans discussing about number bases. It will be machines, computers, roboters. I am very convinced that they will perfectly well speak with each other.

 

[Mark44]

I mean, there are nine numerals.

There are ten digits in base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Similarly, in base-2 (binary), there are two digits; in base-3 (ternary or trinary) there are three digits, in base-16 (hex or hexadecimal), there are 16 digits - the ten digits from decimal plus six more from A, B, C, D, E, and F.

(Corrected)In base-X, the digits run from 0 through X - 1.

 

[Mark44]

Binary has the disadvantage of being a long notation but it is finally a very logical notation.

Binary notation is exactly the same as ternary, octal, decimal, hex, base-32, base-64, etc. A number written as $d_4d_3d_2d_1d_0$ in base-B means $d_4 \times B^4 + d_3 \times B^3 + d_2 \times B^2 + d_1 \times B^1 + d_0 \times B^0$.

Whatever the base happens to be, assuming that it is an integer greater than 1.

Good one. PS did you note that 2019 happened a lot?

As far as I know, 2019 happened only once, back about two years ago. Or perhaps what you meant was "a lot happened in 2019."

 

[Jarvis323]

Aparently Pythagorus's geometry cult thought that 10 was the perfect complete number because 1+2+3+4=10.

Also, apparently some people on the internet have been convinced that using a different numeric base, like 7, would enable new mathematics that could unlock the mysteries of the universe and give us free energy technology, or allow us to transcend into interdimentional beings.

Personally I'm a little bit skeptical.

 

[cmb]

There are ten digits in base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Similarly, in base-2 (binary), there are two digits; in base-3 (ternary or trinary) there are three digits, in base-16 (hex or hexadecimal), there are 16 digits - the ten digits from decimal plus six more from A, B, C, D, E, and F.

In base-X, the digits run from 0 through X - 1.

Yes, of course, but I'm saying I don't see how '0' is a digit that could derive from finger counting.

Count zero on your fingers, then work up. How many do you count?

To get '9' on your last finger you have to fold your first finger and count "zero!". Then hold two fingers down and count "one", three and count "two", etc..

So the Arabic numerals do not match up with fingers. Also, there was no zero in the numeral set in the western world from the time it emerged here (late first millennium, zero appearing mid 2nd millennium).

If '0' was one of the intrinsic digits of the Arabic notation, then '0' would have appeared in Gerbert's original texts where he first described the notation in the west, and it didn't. Apices also wrote on the subject in the West in the 12th century and '0' still hadn't appeared.

I am not at all in a disagreement that our number systems seem to have stemmed from finger counting, but it must have been a multi-stage step. First, develop symbols for number 10 (e.g. 'X' in Roman), then replace it with a positional notation and 'one lot of ten'. So at least a two step process past finger counting.

The Hindu mathematicians had a dot as a place holder from the 7th century which emerged as '0'.

If you count "one" as your first finger bent you get 10 on the last finger. '10' isn't a digit.

I think the only way this could have emerged is that there were representations of numbers-to-fingers and singular symbols for '10' on the last finger (viz. "X" in Roman") so that last finger gets a symbol. After counting "1 to 10" with symbols for a few thousand years, the Indian/Arabic folks figured out that "0 to 9" made more sense.

Fact is, all of the Egyptian, Chinese (rod and later Suzhou), Roman number systems were base 10 and those were also absent '0'. These seem to have done so independently of each other. So the 'because of the number of fingers' argument is pretty solid, difficult to see an alternative explanation.

 

[phinds]

There are ten digits in base-10

In base-X, there are X - 1 digits.

Uh ... ? So base 10 has 10 digits, except that it has 10-1 = 9 digits?

 

[Mark44]

Uh ... ? So base 10 has 10 digits, except that it has 10-1 = 9 digits?

Ulp... What I meant but erred in saying is that in base X, the digits run from 0 through X - 1. I'll fix what I wrote.

 

[phinds]

Ulp... What I meant but erred in saying is that in base X, the digits run from 0 through X - 1. I'll fix what I wrote.

Yeah, I figured that's what you meant. I was just ragging you

 

[jack action]

Of course feel free to pack them together to octs or hex numbers

If this is acceptable, why isn't it acceptable to pack them in decimal numbers?

I'm not saying you are wrong, I just fail to see your point. How does it "make much more sense" to you?

Some have mentioned the alphabet as well. Following your logic, we should also use a binary alphabet over our base-26 system (not counting capital letters and punctuations). That is what our computers do, isn't it? they translate all of these letters in a series of 0 and 1. Imagine reading your favorite masterpiece written in ASCII encoding, where even a space is considered a character (which is logical, isn't it?). One might argue that it is "better", but I'm curious to hear the argumentation explaining how it is better.

 

[cmb]

There are ten digits in base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Similarly, in base-2 (binary), there are two digits; in base-3 (ternary or trinary) there are three digits, in base-16 (hex or hexadecimal), there are 16 digits - the ten digits from decimal plus six more from A, B, C, D, E, and F.

(Corrected)In base-X, the digits run from 0 through X - 1.

On the semantic point, I'm unconvinced 0 is a digit.

If I hold up my 10 digits, which one is the 0th?

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members. Is the "10" in "base 10" a cardinal or an ordinal number?

I don't know, I think it is over-doing it to try to make semantic sense of why we do it this way. Just my POV. We all understand it, why try to over-describe it?

 

[fresh_42]

On the semantic point, I'm unconvinced 0 is a digit.

Well, good that this is irrelevant. 0 is certainly a digit. One may discuss whether it is a natural number, or which I think, a discovery, but it is part of the alphabet and therewith a digit.

If I hold up my 10 digits, which one is the 0th?

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members. Is the "10" in "base 10" a cardinal or an ordinal number?

I don't know, I think it is over-doing it to try to make semantic sense of why we do it this way. Just my POV. We all understand it, why try to over-describe it?

Chomsky shed some light on such discussions. It isn't necessary anymore to babble.

 

[cmb]

Well, good that this is irrelevant. 0 is certainly a digit. One may discuss whether it is a natural number, or which I think, a discovery, but it is part of the alphabet and therewith a digit.

I'd still like someone to address my previous point directly.

"Digit" is latin for "finger".

If you write each of (0 to 9) digits in pen on your fingers, which one do you write "0" on, and which one gets "9"?

I'm sorry but it is pretty bonkers to hold up one's hands, palms up, then bend ones right hand thumb and count 'zero!' then working along to the left hand thumb and say 'nine! all done!'. Just doesn't make much sense to me, but I am content to disagree on that.

Simply, a digit "1" in the second column represents the number ten. The "0" in the first column represents nothing at all, it is just a place holder to show where the second column isn't that column.

 

[Mark44]

On the semantic point, I'm unconvinced 0 is a digit.

I agree with @fresh_42 that 0 is definitely a digit.

If I hold up my 10 digits, which one is the 0th?

Here you are confusing two meanings of the word "digit," one of which refers to numbers while the other has an anatomical meaning.

From the Merriam-Webster dictionary (https://www.merriam-webster.com/dictionary/digit) with emphasis added by me

1a: any of the Arabic numerals 1 to 9 and usually the symbol 0

3: any of the divisions in which the limbs of most vertebrates terminate, which are typically five in number but may be reduced (as in the horse), and which typically have a series of phalanges bearing a nail, claw, or hoof at the tip

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members.

Yes, but so what? The number of digits in any base is always one more than the largest digit; e.g. in binary the number of digits is 2, but the largest digit is 1; in octal, the number of digits is 8, but he largest octal digit is 7, and so on.

Is the "10" in "base 10" a cardinal or an ordinal number?

Cardinal

 

[fresh_42]

I'd still like someone to address my previous point directly.

"Digit" is latin for "finger".

Not very convincing, if it dates back to only 1640.

digit (n.)
late 14c., "numeral below 10," from Latin digitus "finger or toe" (also with secondary meanings relating to counting and numerals), considered to be related to dicere "to say, speak" (from PIE root *deik- "to show," also "pronounce solemnly"). The numerical sense is because numerals under 10 were counted on fingers. The "finger or toe" sense in English is attested from 1640s.

https://www.etymonline.com/search?q=digit

Nevertheless, it doesn't matter where it stems from. Its current usage is relevant, especially if binary is the subject. As such it is an element of the alphabet we use to express calculations, ergo a letter. Digit is just another word for a certain kind of letter.

 

[jbriggs444]

On the semantic point, I'm unconvinced 0 is a digit.

If I hold up my 10 digits, which one is the 0th?

This seems to go to the distinction between cardinal numbers (how many) and ordinal numbers (which one).

As much as anything, it is an accident of language that we a use the same set of numbers for each and that we denote the first cardinal by zero (no objects) and the first ordinal by one (the first object).

It is not always so. In many computer languages, the first element of an array is at index zero. An array with n positions (cardinal number n) has a last position denoted by n-1 (ordinal number n-1).

Note that if you have a set of ten objects and if want to have a numbering system to be able to express the cardinality of any possible subset then you need at least eleven cardinalities.

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members. Is the "10" in "base 10" a cardinal or an ordinal number?

It is a cardinal. The cardinality of the set of digits.

I don't know, I think it is over-doing it to try to make semantic sense of why we do it this way. Just my POV. We all understand it, why try to over-describe it?

Wait, I thought you were the one trying to over-describe it.

 

[cmb]

Wait, I thought you were the one trying to over-describe it.

I was just looking at the different possibilities to see if there is one that offers the simplest explanation that is more self-evident than others, but I think there isn't and we're looking at naming conventions, so no clear answer possible from my POV.

 

[jack action]

I'm loving this nine or ten digits discussion! Let me add fuel to that fire!

So in a base-1 numeral system, following the same logic of other numeral systems, do we consider counting this way:

0, 00, 000, 0000, 00000, ...

But if 0 is nothing, isn't a series of 0 still nothing?

Or this way:

1, 11, 111, 1111, 11111, ...

Still the same problem from a different point of view: Can 1 represent nothing? Or is the absence of a 1 enough to represent 0?

Or this way:

0, 1, 11, 111, 1111, 11111, ...

But then we would be using 2 digits in a base-1 system? Or is it a 1-digit system + a zero?

And still with 2 digits, there is unary code :

0, 10, 110, 1110, 11110, 111110, ...

alternative unary code :

1, 01, 001, 0001, 00001, 000001, ...

So in a unary numeral system, do we consider:

• no zeros;
• a single zero;
• only zeros;
• a mix of both for every number?

 

[Frabjous]

A compromise. Since quantum computers are the future, let‘s just used a counting system based on a superposition all of the proposed bases. I’ll leave the derivation as an exercise to the OP.