If somehow anyone from Numberphile is watching this video please know that me using your same colours/art style is simply an ode to your wonderful maths content. I wasn't in anyway trying to trick people into thinking this was one of your videos. ❤
@@Brennende_Rose that hurts me because that just looks like 2.3 Then again I'm aware a lot of European nations use a comma instead of a period to denote decimals so maybe it isn't as confusing over there.
There is another symbol for multiplication that you didn't mention. The asterisk "*" is used in most computer language for performing multiplication. Division is usually specified by the forward slash "/".
As far as I'm aware, the asterisk is actually just a substitute for the ✕, not an actual mathematical symbol for multiplication (rather something different in geometry that eludes me right now). And the / for division - ou mean like at 5:51? :)
* and / were chosen because they were part of the original ASCII symbols, and were easily accessible on many keyboards. * very roughly approximates the × symbol, and a / is similar to how fractions are written with a line. (in fact it's become common to use the / symbol for plain text fractions)
@@CoolAsFreya I'm not so sure if the / really comes from computers originally (typewriters maybe). My grandparents always wrote vulgar fractions in this "diagonal" way. Those fractions' Unicode symbols whose origins date back quite a bit also look like that (½, ¼, ...). But that has an obvious reason - it fits in single line in normal writing. Hence the slash was mentioned in the video and the asterisk wasn't. The former has precedence, the latter is native to programming.
"/" is _slash_, not _forward slash_ . I have been involved in computing for over 50 years, and it has always been so. The asterisk is used in computer languages as the × symbol was not available on many, if any, keyboards.
As a mathematics student I'd have loved to see you cover some more symbols like: ∈ (meaning element of a set) ⊆ (subset of a set) ∫ (integral) Σ (sum over a set or index) {x,y} (denoting a set of x and y) ∅ (the empty set) f(x) (meaning function of x) ℕ,ℤ,ℚ,ℝ,ℂ being different sets of numbers (of course their history is quite obvious) < (less than) parentheses ∪ and ∩ (union and intersection of sets) x² (meaning x squared) √x (meaning square root)
@@scoutgaming737 Why did you call all the set symbols "nonsensical"? ∈ is a rounded E for "element of", ⊆ is a rounded less-than-or-equal sign, ∪ is a U for "union", ∩ is the upside down version, because union and intersection have a complementary relationship. They are also rounded versions of the boolean operators ∨ ("or" - a V for Latin "vel") and ∧ ("and"), because set operations and boolean algebra are closely related.
The integral sign is just a long s and stands for summation (since integration is sort of an infinite summation). The sign for regular summation is a capital sigma for the same reason.
That text in Middle English more probably says "I will set, as I do often in work use, a pair of parallels, of Gemini (or twin) lines if one length, thus: === ..." Here "set" means "write" as in "set pen to paper". Gemini is the constellation representing the twins Caster and Polux from Greek mythology, so "Gemini lines" are a pair of lines.
I (Dutch) was taught from middle school onward to use • instead of ×, because "× is too much like x and x will be used a lot in math", so now • is more natural than × to me.
I was going to comment the same thing, but in Brazil: in arithmetics, we use × until 6th grade *that is primary school) but from then on, • or even nothing (if you have parenthesis, 5 (2+3) is the same thing as 5 • (2+3), no symbol needed).
you write you X's in maths like a normal x? not the curly x that looks completely different to the multiplication symbol? in england we write some letter in algebra differently like 𝑥 so it doesnt get confusing when theres a × next to an 𝑥 cause they look so different
Great video, just a few little inaccuracies I saw 1. In maths above high-school level IME the dot is quite a bit more common than the cross, and the most common notation by far is just omitting the symbol entirely. 2. at 7:57 you say the original text Robert Recorde wrote is Old English, but I think it would be better to clarify that it's early Modern English ("Shakespeare's English"); Old English is properly the form of the language spoken upto the 11th century or so.
Memory serving, I was taught that the percent sign was just a corruption of the number 100. Given how the per-1000 sign is just a percent sign with an extra zero on the bottom, I think this theory either holds water altogether or at least held water with the inventor of the per-1000 sign.
Except for algebra and most scientific fields, particularly where letters (including from foreign alphabets) represent a single magnitude or ratio: e for the base of the natural logarithm, i (j to the sparkies) for √(-1), π for the ratio of the circumference of a circle to its diameter, τ for the circumference of a circle to its radius, φ for the ratio of a golden rectangle to its width, ℵ₀ for the number of elements in several sets of numbers, and ω for the set of all finite ordinals.
Way back in elementary school I learned that the divide symbol is a line literally dividing a group and that the percent symbol is simply a rearrangement of the digits in 100.
I was taught something very similar and I have seen old written depictions showing one zero or no zero in the top and 100 or double zeros on rhe bottom ... even a few old adding machines and calculators with a key labeled as such.
In Finland the colon (:) is used for division but that is only used in early classes, later a line is typically used or a slash. Multiplication is typically a dot though "×" has also been used but I think it somewhat old fashioned.
In relatively simple terms, the origins of the names of the operators are: "plus" from Latin "plūs" meaning "more" "add" from Latin "addō" meaning "give unto" from "ad" meaning "to" + "dō" meaning "give" "minus" from Latin "minus", neuter form of "minor", comoparative form of "parvus" meaning "small" or "little" "subtract" from Latin "subtractus" perfect passive partisiple of "subtrahō" meaning "I draw from beneath" or "withdraw" or "remove", from sub meaning "under" and "trahō" meaning "I draw" "multiply" from Latin "multiplicō" from "multi" meaning "many" + "plicō" meaning "I fold" "divide" from Latin "dīvidere" meaning "to divide", from "dis" meaning "two" and "vidō" meaning "to separate"
William Oughthred is primarily remembered for developing the slide rule which is based on logarithms. Logarithms convert multiplication to addition, so it makes sense that he would want a symbol for multiplication.
SO interesting! would definitely be interested in a series on this about more complicated math symbols like partial derivatives, gradients, composite functions, and matrices. love this stuff!
@8:00 Old English? I don't think so. That sentence is quite clearly Modern English, as (a) you can easily read it (aside from the interesting spelling choices) and (b) people had been speaking Modern English for quite some time by 1557. Old English stopped being a thing by around 1100. And Middle English had transitioned into Modern English by around 1500.
I went to school in Germany. We were discouraged from using × and ÷ even though they were on the pocket calculator; when I used them the teacher marked it red and asked whether I meant x and whether ÷ meant : or −. Since I learned programming I usually use /. You mentioned that Leibniz introduced the central dot ·, that's what we used for multiplication. I like the simple symbolism of the signs < and > with a big end to the larger number and a little point to the smaller value.
"letters don't have much of a role in math" bruh we use so many letters in math that we literally ran out and started using letters from the Greek and Hebrew alphabets. Almost every letter of the Latin and Greek alphabets is used rather commonly, Hebrew is a bit more rare. And among mathematicians, the centered dot is certainly more common than the cross for multiplication, although in many contexts they are both used with different meanings.
Have you ever known the “therefore” (∴) symbol? This one might be a little obscure, but it’s used often in my high school math class. Perhaps you could make the second part of the video
@@MatsUtterheim And per ten thousand has three zeroes: ‱. Although these two symbols seem to have been derived from the percent sign, since historical evidence shows the "o" of the % symbol originating from the -o suffix in Italian, and not from the number zero.
It also looks like a variation of the division sign (similar to the approx. equal vs equal) where it's a specific type of division, that being X out of 100.
9:30 and the symbols have spawned even more symbols. The percent symbol has spawned a 'per thousand' symbol, and even a 'per ten thousand' symbol, made by adding one or two additional 'small zeroes' to the right side. I'll try to approximate them here: °/oo (per thousand) °/ooo (per ten thousand)
Ahhhh, Patrick, with your accent and your, I guess British tradition of keeping the 's' around after shortening Mathmatics, your pronunciation of Math's (my phone just changed the plural of math I typed to the possessive, see even Samsung agrees) sounds like MAFFS the longer I listen to you. Love your channel
a sequel to this video that explains the square root symbol the calculus symbols and the use of various Greek letters to mean different functions would be awesome
But what about > and < being derived from the = sign too? Instead of the parallel lines showing equality, the convergence/divergence of the line segments implies “is it smallest here, but on the other side of symbol it is at its largest.” So, going small to big, we think
"Though perhaps the most widely used sign is the equal sign. It's probably the most used as it appears in all sums regardless whether it's an addition, subtraction, multiplication, or division." The word "sum" refers specifically to addition. The words for the other ones are "difference," "product," and "quotient."
@@gracchus7782 No, it's just widely used slang. A "big equation" might well be called a "big sum" but anyone who works with maths would almost always say equation.
It's incredible to think about how much more advanced we would be (probably) if the ancient Greeks had had symbols for numbers and functions. Just to do something simple, they had to use words, as in "twoplustwoequalsfour", with not even spaces between the words. To do something complicated it would take a genius like Archimedes and that was still an absolute nightmare.
This is especially interesting since here in Germany, when being taught English in school, it's more likely to learn how to read Shakespeare than to learn the math symbols. I didn't know how to pronounce or spell a lot of basic math symbols in English until my second year of studying computer science. You can still read many scientific papers with all kinds of math symbols, but as they are never written as words, you have to read them in your mother language in your head. There are some cases where this matters a lot. For example I was very surprised when I learned that x over y and its direct translation "x über y" are completely different operations.
@@klikkolee"x choose y", the binomial coefficient. It is used to find the number of ways of selecting y different things from a set of x different things.
Too bad that Shakespeare never wrote a play about math. You know, like A Midsummer Night's Extracting A Square Root By The Long-Division-Looking Algorithm That Nobody Ever Remembers Even The Ones Who Scored Perfectly On That Test..
3:50 Though if you're a programmer, while "2 + 3" usually equals 5, "2 & 3" sometimes equals 2. The & symbol is used in some programming languages as the "bitwise and" operator. Thus, since 2 and 3 are 0010 and 0011 in binary, respectively, only the bits that are 1 in both numbers will be 1 in the result.
@@Anonymous-df8it No, that would be bitwise xor (which I'm not sure of a common symbol for off the top of my head). "&" is more like binary addition, but you keep only the carry bit, and keep it in the same spot instead of carrying it to the left.
@@Anonymous-df8it Nah, that's ok. It's probably good to keep the clarification, in case anyone else who comes across this thread has the same thought. So I don't think you need to delete either comment.
I grew up using the dot for multiplication, likely because I grew up in Germany, where Leibniz is from. As a kid I was always wondering why many calculators used the “weird” ✖️-symbol instead of the “normal” dot 😅
In continental Europe the central dot for a multiplication sign is universal. The X is rarely used at all, because that just brings confusion when the X is used for the common variable. I also remember the division sign in school being mostly this sign : but also sometimes it was an L shape of sorts, with the horizontal line of the L being elongated and going all the way underneath the denominator. I've never seen the horizontal line with the two dots being used anywhere at all for the division sign outside of the button on the calculator.
"Robert Recorde (c. 1510 - 1558) [...] because he actually wrote in Old English [...] This creation came into being in 1557". It is at least five centuries too late to be Old English. It would not even count as Middle English. From the look of it is Early Modern English, which would be expected in the 16th century. It should be fairly intelligible to most native English speakers without much trouble.
I had always thought of Division & percent to be a reference to the fact that they are fractions. As the idea of a fraction being. Whatever (dot) over (line) whatever (dot) [ ÷ ] Also that the percent sign was similar. As it is sometimes a single 0 over a double 00 [‰]
In real math, multiplication is always a dot, you last see the cross in primary school, maybe high school in some places, and then it's never used this way again. Cross even gets an alternative meaning, such as the cross product of vectors. Same goes for division sign... division is always represented with a fraction (or a slash). The obelus is only seen on a calculator button. This also solves the problem of order of operations if division is written in-line with the rest of the operations.
Equals (=) is two parallel lines of equal length. This can then be compared to the less than () symbols, where the smaller side of the symbol is close to the smaller value of the comparison, and the larger side of the symbol is close to the larger value of the comparison. But these symbol came along after equals.
When dealing with vectors, the dot product and cross product are two distinct multiplication-like operations which use the dot and the ×, respectively.
The origin of the percent sign is plausible because of a similar case with an abbreviation that has come into being in recent decades. In computer programming, translation of on screen text is called internationalization. Some programmer a few decades ago got tired of writing that and abbreviated it i18n. That’s because internationalization starts with an ‘i’, is followed by 18 letters, and ends with an ‘n’. The abbreviation stuck. Now the whole software industry does that.
@@Anonymous-df8it I’ve never heard an explanation. My guess is because the same general process deals with regional differences within the same language. For example, there are separate language codes for US and UK English so that systems know which spell checker and keyboard layout to use.
Such symbols have been used in certain fields mainly as measurements of concentration with "ppm" (parts per million) and "ppb" (parts per billion) are used to indicate trace amounts.
@@JamesDavy2009 Since "billion" has two meanings (10^9 and 10^12), does "ppb" inherit both depending on context? Either way, it would be quite confusing (if "ppb" inherits both meanings, all the confusion over "billion" would be inherited; otherwise, it would be confusing if you used the non-inherited definition of "billion", since they'd no longer be each other's reciprocal) Also, I'm curious if different languages use different initialisms
@@Anonymous-df8it Two languages I know of: Chinese and Japanese group numbers by the myriad (1 million = 100 myriad) so they may have equivalent terms for "parts per milliard (10^8)" and "parts per billiard (10^12)". Most western countries are increasingly interpreting "billion" in the short scale context (equivalency: 10^9).
@@JamesDavy2009 A milliard is 10^9, not 10^8 and a billiard is 10^15, not 10^12. Also, what about languages that use the long or short scale, but yet still have different words for "parts" and/or "per"?
It's wild to me how much "×" and "÷" are used in teaching arithmetic, because they almost totally disappear in more advanced math. "÷" is almost always replaced with "/" if it's got to stay on one line, or stacked if you have room to spare. While "⋅" is preferred to "×", multiplication is also so common that it usually doesn't use a symbol. Instead of "2⋅a⋅b," you'll always see "2ab." Multiplied numbers usually get solved, but I swear you're more likely to see "2(3)" than "2⋅3."
The • for multiplication is definetely considered the "proper" way to do multiplication equations here in highschool and onwards. It's pretty quick to write too! But yeah × is used a lot too so both are natural
once you get to algebra and above the divide by symbol is rarely used. the fraction (/) is far more clear and intuitive in an equation and eliminates any ambiguity with orders of operations.
"Imagine if you could only add to things" In computer hardware, subtraction is actually done via addition: invert the bits of the second value, and add with a carry of 1 in.
07:57, a more accurate translation of the old english text would be: Because I write it so often when I work, I will use two parallell lines of twin length. Because no two things can be more equal.
The legs “waking toward” the answer actually reminded me of greater than/less than. When I was a kid, the way I came up to remember which was which was that the symbol was a mouth “talking to” the bigger number.
Actually, I come from Belgium and we use the dot a lot for multiplication especially in secondary education. Also, it's doesn't really matter if you can't see the dot because multiplication can most of the time not be written at all 😄
in my schooling, the × and ÷ symbols were only really used in elementary school. Even though i rarely do math these days if i ever write down an equation , 100% of the time i'm using the dot for multiplication and a fraction notation for division. it makes a million times more sense and infact the multiplication and division signs introduce a level of ambiguity that is completely unsuited for doing math
In Finnish school (I started around mid 2000s) we never used ×. Not even any teacher had it in personal use. Because of this I've always seen it as a foreign thing limited to some calculator apps.
Will there be a sequel to this topic covering other math symbols such as the Square Root, less than ()? Also, because I heard you say "a bit of a faff," could you explain British phrases that sound odd to non-Brits, such as gutted, chuffed, and wind-up
interesting that this etymology of the percent symbol has a letter 'o' underneath, where the per mill symbol (‰) must've been based on the lower number looking like a 0.
My personal theory about the % symbol (which I long assumed would never be more than a theory) is rooted in its meaning. Since "percent" literally means "of one hundred", it was probably originally notated as "/100". Of course, two near-vertical lines written rapidly in succession would cause legibility issues, the 1 was eventually dropped to form "/00". Similar to the evolution of the ampersand (&), this gradually morphed into "0/0"; making the relationship with % pretty darned obvious.
The × and • symbols actually have slightly different meanings in physics. It's hard to explain fully (perhaps someone can help?) But it has to do with combining tensors. Essentially, the × or "cross product" is just the raw multiplication of all of the numbers, while the • or "dot product" takes into account all of the magnitudes and directions of all the different tensor values in relation to one another to determine a net vector
The dot for multiplication would cause confusion as it could be confused with the decimal point (in the English speaking word) or the thousands seperator as used in Europe.
Ways if saying + - Plus and Minus - common language Add(ition) and Subtract(ion) - more "professional" or formal Positive and Negative - the value itself
in computing, % is used as the modulo (often shortend to just "mod") operator symbol i.e. the remainder after integer division e.g. 11 % 2 = 1. There isn't a single symbol (that I am aware of, at least) to represent integer division, but in the programming languages I know best, it is "//" e.g. 11 // 2 = 5. It's just the counterpart of modulo
@@Anonymous-df8it because computers are deterministic and do not have the possibility of misinterpreting the input. To the computer, % is unambiguously the modulo operator and percentage is not an operation unto itself. To take the percent of something you would have to do a multiplication and division operation (* and / respectively)
I have always thought that the division symbol is a line dividing two dots. Two divided in two, is one on either side. That's probably wrong, but I thought of in second grade (age 6)
In computer languages, the double asterisk (**) or the upwards pointing arrow (↑) are used to indicate exponentiation (raising to a power). The older computers simply had no way of superscripting numbers the way that this was traditionally done. Frankly, it’s awkward even on modern computers and is very difficult to see.
The latter symbol is used in Knuth's up-arrow notation and is instrumental in the calculation of Graham's Number. Nowadays we use the carat (^) symbol in lieu of the up arrow for typed exponentiation.
Using the dot to mean "multplied by" is a bit confusing. It means you have to ask, 'Does 4•7 mean "four times seven" or "four point seven" (i.e. four and seven tenths)?'
i disagree with one of your statements. 5:28 i think cdot (center dot) is actually closer to perfect (as long as you have good pen and/or vision or attention), as someone that use nothing or cdot as multiplication. i only use × for cross product of vectors. i would argue we should ditch × as multiplication sign at all education levels, but cdot can go unnoticed as you said in video, so i think every teacher should teach multiplication sign as tiny cross that is centered on the grid/line, and encourage kids to make the cross tinier each year (or it might be how it is being taught in some places rn, idk)
saying that letters don't play much role and that the multiplication dot is unpopular is very alien to me. "Math with letters" was a primary school thing for me, and when you look at math that has anything to do with the real world, it's full of letters. Keeping the multiplication cross after primary school math is just not going to work. Multiplication is also extremely common in applied math, so a more condensed notation is strongly favored. When possible, multiplication is indicated by just putting the factors next to each other, but when that's impractical or ambiguous, it's *always* the dot.
Yep. They're pretty easy to distinguish in any font meant for math, but you're generally taught that the multiplication sign is just a floating x so it's hard with most people's handwriting. But I'm definitely also not the only mathematician who's deliberately changed my handwriting to make my math notes easier to read...
Kind of surprised you decided not to mention two other quite popular means for annotating multiplication -- the asterisk (*) and of course, just simple parentheses between neighboring terms (both of which are almost always more popular than the classic × in typed rather than handwritten math) Also, a percent is not "normally out of 100," as you said; that's the literal _definition_ of "percent." It's *always* with respect to 100.
What you refer to is the triple bar used mostly in logical mathematics to denote that x is logically equivalent to y and outside of that context it means x is identical to y.
It can also mean congruence. Two expressions are congruent if they equal each other no matter what the variables are set to. Two numbers are congruent modulo a third if they yield the same remainder after dividing by the third
Please don't use & instead of +. It is already in use in programming as a completely different operation, combining the binary representation of numbers, making a bit 1, only if it is 1 in one of the numbers. So 2 & 3 = 2 (0010 & 0011 = 0010).
But "mathematics" is not usually used as plural in English. It's most often used as a singular noun in modern usage. (e.g., "Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes." - en.wikipedia.org) French does still usually treat it as a plural noun: « Les mathématiques sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations, etc. » - fr.wikipedia.org
I mean, 'mathematics' behaves grammatically in the singular (since it's uncountable; one does not go "one mathematic, two mathematics..."). Also, my computer hates it if I spell it like that... ...I'd agree with you on the spelling however in writing or in speech, but I'd rather just not abbreviate it when typed, since I don't like the computer complaining
If somehow anyone from Numberphile is watching this video please know that me using your same colours/art style is simply an ode to your wonderful maths content. I wasn't in anyway trying to trick people into thinking this was one of your videos. ❤
A perfect choice! ❤
It's also a lovely hommage that you used "mathS" instead of "math"... :-)
@@markstyles1246That's not an homage, that's how we say it in the UK
@@davidcarney1533 Did I need a "/s" not just a "... :-)"?
@@markstyles1246 In this instance, yes 😂
In linear algebra, × (cross product) and ⋅ (dot product) have distinct meanings.
Not to mention the totally different meaning of × in set theory!
In German we use the dot just for ordinary equations, such as 2*3=6
@@Brennende_Rose Same in Denmark, but that makes sense since it's just a dot product of 1 dimensional vectors
@@Brennende_Rose Also, in Germany we use a simple colon for division, as in 6:2=3.
@@Brennende_Rose that hurts me because that just looks like 2.3 Then again I'm aware a lot of European nations use a comma instead of a period to denote decimals so maybe it isn't as confusing over there.
There is another symbol for multiplication that you didn't mention. The asterisk "*" is used in most computer language for performing multiplication. Division is usually specified by the forward slash "/".
As far as I'm aware, the asterisk is actually just a substitute for the ✕, not an actual mathematical symbol for multiplication (rather something different in geometry that eludes me right now). And the / for division - ou mean like at 5:51? :)
* and / were chosen because they were part of the original ASCII symbols, and were easily accessible on many keyboards. * very roughly approximates the × symbol, and a / is similar to how fractions are written with a line. (in fact it's become common to use the / symbol for plain text fractions)
@@CoolAsFreya I'm not so sure if the / really comes from computers originally (typewriters maybe). My grandparents always wrote vulgar fractions in this "diagonal" way. Those fractions' Unicode symbols whose origins date back quite a bit also look like that (½, ¼, ...). But that has an obvious reason - it fits in single line in normal writing. Hence the slash was mentioned in the video and the asterisk wasn't. The former has precedence, the latter is native to programming.
"/" is _slash_, not _forward slash_ . I have been involved in computing for over 50 years, and it has always been so.
The asterisk is used in computer languages as the × symbol was not available on many, if any, keyboards.
@@frogandspannerPeople say forward slash to avoid answering "which slash?"
As a mathematics student I'd have loved to see you cover some more symbols like:
∈ (meaning element of a set)
⊆ (subset of a set)
∫ (integral)
Σ (sum over a set or index)
{x,y} (denoting a set of x and y)
∅ (the empty set)
f(x) (meaning function of x)
ℕ,ℤ,ℚ,ℝ,ℂ being different sets of numbers (of course their history is quite obvious)
< (less than)
parentheses
∪ and ∩ (union and intersection of sets)
x² (meaning x squared)
√x (meaning square root)
@@scoutgaming737I think that stretched out s is called an “esh”
@@tlachers the Esh name in phonology came after the symbol's creation for calculus
@@scoutgaming737 Why did you call all the set symbols "nonsensical"? ∈ is a rounded E for "element of", ⊆ is a rounded less-than-or-equal sign, ∪ is a U for "union", ∩ is the upside down version, because union and intersection have a complementary relationship. They are also rounded versions of the boolean operators ∨ ("or" - a V for Latin "vel") and ∧ ("and"), because set operations and boolean algebra are closely related.
@@scoutgaming737 The element symbol comes from the letter E, union from the letter U and intersect from the opposite of union
@@scoutgaming737 Maybe instead of saying "don't know," you could ask if someone else knows and we can have a more fun and more engaging conversation.
The integral sign is just a long s and stands for summation (since integration is sort of an infinite summation). The sign for regular summation is a capital sigma for the same reason.
That text in Middle English more probably says "I will set, as I do often in work use, a pair of parallels, of Gemini (or twin) lines if one length, thus: === ..."
Here "set" means "write" as in "set pen to paper".
Gemini is the constellation representing the twins Caster and Polux from Greek mythology, so "Gemini lines" are a pair of lines.
I (Dutch) was taught from middle school onward to use • instead of ×, because "× is too much like x and x will be used a lot in math", so now • is more natural than × to me.
I always hated that × looked so much like an X.
I associate the interpunct with Catalan, not with mathematics.
I'm from the US I was taught the same thing in 5th grade
I was going to comment the same thing, but in Brazil: in arithmetics, we use × until 6th grade *that is primary school) but from then on, • or even nothing (if you have parenthesis, 5 (2+3) is the same thing as 5 • (2+3), no symbol needed).
you write you X's in maths like a normal x? not the curly x that looks completely different to the multiplication symbol? in england we write some letter in algebra differently like 𝑥 so it doesnt get confusing when theres a × next to an 𝑥 cause they look so different
I love that the approximately equal symbol ≈ is related to both the equals symbol = and the tilde symbol ~ which is often used to mean "approximately"
The tilde became used in this way because it is found on standard keyboards, and the standard approximation symbol ≈ isn't.
The tilde can also mean proportional as in a~b meaning a is directly proportional to b.
Great video, just a few little inaccuracies I saw
1. In maths above high-school level IME the dot is quite a bit more common than the cross, and the most common notation by far is just omitting the symbol entirely.
2. at 7:57 you say the original text Robert Recorde wrote is Old English, but I think it would be better to clarify that it's early Modern English ("Shakespeare's English"); Old English is properly the form of the language spoken upto the 11th century or so.
1 is true but you have to write the symbol if multiplying two numbers
I always assumed the division symbol was an abstracted pictograph of a fraction with the two dots being placeholders for the two numbers
I've pictured it that way too.
Indeed one can add zeroes to the quotient part of the % to indicate per thousand (‰) or per ten thousand (‱).
i was told this by multiple teachers growing up, i thought it was common knowledge
Memory serving, I was taught that the percent sign was just a corruption of the number 100. Given how the per-1000 sign is just a percent sign with an extra zero on the bottom, I think this theory either holds water altogether or at least held water with the inventor of the per-1000 sign.
"By and large, letters don't really play much of a role in maths"
me, an engineer:
Except for algebra and most scientific fields, particularly where letters (including from foreign alphabets) represent a single magnitude or ratio: e for the base of the natural logarithm, i (j to the sparkies) for √(-1), π for the ratio of the circumference of a circle to its diameter, τ for the circumference of a circle to its radius, φ for the ratio of a golden rectangle to its width, ℵ₀ for the number of elements in several sets of numbers, and ω for the set of all finite ordinals.
I do maths every day. It rarely involves numbers.
Way back in elementary school I learned that the divide symbol is a line literally dividing a group and that the percent symbol is simply a rearrangement of the digits in 100.
I was taught something very similar and I have seen old written depictions showing one zero or no zero in the top and 100 or double zeros on rhe bottom ... even a few old adding machines and calculators with a key labeled as such.
In Finland the colon (:) is used for division but that is only used in early classes, later a line is typically used or a slash. Multiplication is typically a dot though "×" has also been used but I think it somewhat old fashioned.
Cool
In relatively simple terms, the origins of the names of the operators are:
"plus" from Latin "plūs" meaning "more"
"add" from Latin "addō" meaning "give unto" from "ad" meaning "to" + "dō" meaning "give"
"minus" from Latin "minus", neuter form of "minor", comoparative form of "parvus" meaning "small" or "little"
"subtract" from Latin "subtractus" perfect passive partisiple of "subtrahō" meaning "I draw from beneath" or "withdraw" or "remove", from sub meaning "under" and "trahō" meaning "I draw"
"multiply" from Latin "multiplicō" from "multi" meaning "many" + "plicō" meaning "I fold"
"divide" from Latin "dīvidere" meaning "to divide", from "dis" meaning "two" and "vidō" meaning "to separate"
That's what I call profound knowledge. Thanks and respect!
@@_InTheBin Also, the "ad" part of "addo", meaning "to", is directly where we get the word "at" from.
@@scmtuk3662 Thanks, I know, historic linguist here. Peace.
William Oughthred is primarily remembered for developing the slide rule which is based on logarithms. Logarithms convert multiplication to addition, so it makes sense that he would want a symbol for multiplication.
They are also one inversion of the exponentiation operation, the other one involving the radical sign.
SO interesting! would definitely be interested in a series on this about more complicated math symbols like partial derivatives, gradients, composite functions, and matrices. love this stuff!
@8:00 Old English? I don't think so. That sentence is quite clearly Modern English, as (a) you can easily read it (aside from the interesting spelling choices) and (b) people had been speaking Modern English for quite some time by 1557. Old English stopped being a thing by around 1100. And Middle English had transitioned into Modern English by around 1500.
I went to school in Germany. We were discouraged from using × and ÷ even though they were on the pocket calculator; when I used them the teacher marked it red and asked whether I meant x and whether ÷ meant : or −. Since I learned programming I usually use /. You mentioned that Leibniz introduced the central dot ·, that's what we used for multiplication.
I like the simple symbolism of the signs < and > with a big end to the larger number and a little point to the smaller value.
"letters don't have much of a role in math" bruh we use so many letters in math that we literally ran out and started using letters from the Greek and Hebrew alphabets. Almost every letter of the Latin and Greek alphabets is used rather commonly, Hebrew is a bit more rare. And among mathematicians, the centered dot is certainly more common than the cross for multiplication, although in many contexts they are both used with different meanings.
Have you ever known the “therefore” (∴) symbol? This one might be a little obscure, but it’s used often in my high school math class. Perhaps you could make the second part of the video
% is actually a just 0/0 where the zeroes represent 100
Yeah! And thats why per mille has an extra zero ‰
@@MatsUtterheim And per ten thousand has three zeroes: ‱. Although these two symbols seem to have been derived from the percent sign, since historical evidence shows the "o" of the % symbol originating from the -o suffix in Italian, and not from the number zero.
It also looks like a variation of the division sign (similar to the approx. equal vs equal) where it's a specific type of division, that being X out of 100.
9:30 and the symbols have spawned even more symbols. The percent symbol has spawned a 'per thousand' symbol, and even a 'per ten thousand' symbol, made by adding one or two additional 'small zeroes' to the right side. I'll try to approximate them here: °/oo (per thousand) °/ooo (per ten thousand)
Ahhhh, Patrick, with your accent and your, I guess British tradition of keeping the 's' around after shortening Mathmatics, your pronunciation of Math's (my phone just changed the plural of math I typed to the possessive, see even Samsung agrees) sounds like MAFFS the longer I listen to you. Love your channel
nobody talking about the 5 seconds of silence at the start of the video?
Evidently not. You could make a "Name Explain Explained" video about it.
a sequel to this video that explains the square root symbol the calculus symbols and the use of various Greek letters to mean different functions would be awesome
Some Greek letters represent actual numbers: π, τ (equivalency: 2π) and φ
But what about > and < being derived from the = sign too? Instead of the parallel lines showing equality, the convergence/divergence of the line segments implies “is it smallest here, but on the other side of symbol it is at its largest.” So, going small to big, we think
"Though perhaps the most widely used sign is the equal sign. It's probably the most used as it appears in all sums regardless whether it's an addition, subtraction, multiplication, or division." The word "sum" refers specifically to addition. The words for the other ones are "difference," "product," and "quotient."
I noticed that too. Was wondering if the term "sum" had a wider meaning in British English
They're all sums because every operation can be represented by repeated addition
@@gracchus7782 No, it's just widely used slang. A "big equation" might well be called a "big sum" but anyone who works with maths would almost always say equation.
It's incredible to think about how much more advanced we would be (probably) if the ancient Greeks had had symbols for numbers and functions. Just to do something simple, they had to use words, as in "twoplustwoequalsfour", with not even spaces between the words. To do something complicated it would take a genius like Archimedes and that was still an absolute nightmare.
Euclid was the main greek mathematician, and yes his text was also an "absolute nightmare" to anyone these days.
This is especially interesting since here in Germany, when being taught English in school, it's more likely to learn how to read Shakespeare than to learn the math symbols. I didn't know how to pronounce or spell a lot of basic math symbols in English until my second year of studying computer science. You can still read many scientific papers with all kinds of math symbols, but as they are never written as words, you have to read them in your mother language in your head.
There are some cases where this matters a lot. For example I was very surprised when I learned that x over y and its direct translation "x über y" are completely different operations.
What operation is "x über y"?
@@klikkolee"x choose y", the binomial coefficient. It is used to find the number of ways of selecting y different things from a set of x different things.
Too bad that Shakespeare never wrote a play about math. You know, like A Midsummer Night's Extracting A Square Root By The Long-Division-Looking Algorithm That Nobody Ever Remembers Even The Ones Who Scored Perfectly On That Test..
3:50 Though if you're a programmer, while "2 + 3" usually equals 5, "2 & 3" sometimes equals 2. The & symbol is used in some programming languages as the "bitwise and" operator. Thus, since 2 and 3 are 0010 and 0011 in binary, respectively, only the bits that are 1 in both numbers will be 1 in the result.
So "&" just means binary addition without carrying
@@Anonymous-df8it No, that would be bitwise xor (which I'm not sure of a common symbol for off the top of my head). "&" is more like binary addition, but you keep only the carry bit, and keep it in the same spot instead of carrying it to the left.
@@nedhunter4444 Should I delete my initial reply? Should I delete this one as well? Please answer each question individually
@@Anonymous-df8it Nah, that's ok. It's probably good to keep the clarification, in case anyone else who comes across this thread has the same thought. So I don't think you need to delete either comment.
I grew up using the dot for multiplication, likely because I grew up in Germany, where Leibniz is from. As a kid I was always wondering why many calculators used the “weird” ✖️-symbol instead of the “normal” dot 😅
In continental Europe the central dot for a multiplication sign is universal. The X is rarely used at all, because that just brings confusion when the X is used for the common variable. I also remember the division sign in school being mostly this sign : but also sometimes it was an L shape of sorts, with the horizontal line of the L being elongated and going all the way underneath the denominator. I've never seen the horizontal line with the two dots being used anywhere at all for the division sign outside of the button on the calculator.
"Robert Recorde (c. 1510 - 1558) [...] because he actually wrote in Old English [...] This creation came into being in 1557". It is at least five centuries too late to be Old English. It would not even count as Middle English. From the look of it is Early Modern English, which would be expected in the 16th century. It should be fairly intelligible to most native English speakers without much trouble.
I had always thought of Division & percent to be a reference to the fact that they are fractions.
As the idea of a fraction being. Whatever (dot) over (line) whatever (dot)
[ ÷ ]
Also that the percent sign was similar.
As it is sometimes a single 0 over a double 00 [‰]
I didn't know that the obelus was originally used similarly to the Japanese rice symbol ※.
Actually, ‰ is the per mille sign, which means thousandth.
In real math, multiplication is always a dot, you last see the cross in primary school, maybe high school in some places, and then it's never used this way again. Cross even gets an alternative meaning, such as the cross product of vectors. Same goes for division sign... division is always represented with a fraction (or a slash). The obelus is only seen on a calculator button. This also solves the problem of order of operations if division is written in-line with the rest of the operations.
Division carries the same precedence as multiplication as it's multiplying by the reciprocal of a number.
Equals (=) is two parallel lines of equal length. This can then be compared to the less than () symbols, where the smaller side of the symbol is close to the smaller value of the comparison, and the larger side of the symbol is close to the larger value of the comparison. But these symbol came along after equals.
When dealing with vectors, the dot product and cross product are two distinct multiplication-like operations which use the dot and the ×, respectively.
The origin of the percent sign is plausible because of a similar case with an abbreviation that has come into being in recent decades. In computer programming, translation of on screen text is called internationalization. Some programmer a few decades ago got tired of writing that and abbreviated it i18n. That’s because internationalization starts with an ‘i’, is followed by 18 letters, and ends with an ‘n’. The abbreviation stuck. Now the whole software industry does that.
Why isn't it just called "translation"?
@@Anonymous-df8it I’ve never heard an explanation. My guess is because the same general process deals with regional differences within the same language. For example, there are separate language codes for US and UK English so that systems know which spell checker and keyboard layout to use.
There's also a "per thousand" symbol. Looks like percent % but with two zeros ‰ on the bottom. "Per mille."
Such symbols have been used in certain fields mainly as measurements of concentration with "ppm" (parts per million) and "ppb" (parts per billion) are used to indicate trace amounts.
@@JamesDavy2009 Since "billion" has two meanings (10^9 and 10^12), does "ppb" inherit both depending on context? Either way, it would be quite confusing (if "ppb" inherits both meanings, all the confusion over "billion" would be inherited; otherwise, it would be confusing if you used the non-inherited definition of "billion", since they'd no longer be each other's reciprocal)
Also, I'm curious if different languages use different initialisms
@@Anonymous-df8it Two languages I know of: Chinese and Japanese group numbers by the myriad (1 million = 100 myriad) so they may have equivalent terms for "parts per milliard (10^8)" and "parts per billiard (10^12)". Most western countries are increasingly interpreting "billion" in the short scale context (equivalency: 10^9).
@@JamesDavy2009 A milliard is 10^9, not 10^8 and a billiard is 10^15, not 10^12. Also, what about languages that use the long or short scale, but yet still have different words for "parts" and/or "per"?
It's wild to me how much "×" and "÷" are used in teaching arithmetic, because they almost totally disappear in more advanced math. "÷" is almost always replaced with "/" if it's got to stay on one line, or stacked if you have room to spare. While "⋅" is preferred to "×", multiplication is also so common that it usually doesn't use a symbol. Instead of "2⋅a⋅b," you'll always see "2ab." Multiplied numbers usually get solved, but I swear you're more likely to see "2(3)" than "2⋅3."
Why don't we teach multiplication and division with the standard "2(3)" and "2/3" instead?
Everyoneknows that the multiplication sign was conceived of after a tornado turned every plus-shaped object by 45 degrees. /j /ref
The • for multiplication is definetely considered the "proper" way to do multiplication equations here in highschool and onwards. It's pretty quick to write too! But yeah × is used a lot too so both are natural
The division sign being the oldest of the basic arithmetic symbols was very surprising!
once you get to algebra and above the divide by symbol is rarely used. the fraction (/) is far more clear and intuitive in an equation and eliminates any ambiguity with orders of operations.
"Imagine if you could only add to things"
In computer hardware, subtraction is actually done via addition: invert the bits of the second value, and add with a carry of 1 in.
07:57, a more accurate translation of the old english text would be:
Because I write it so often when I work, I will use two parallell lines of twin length. Because no two things can be more equal.
Two more symbols the more than > sign and the less than sign
I recall it is a larger side and smaller side. Like an = but tilt to show unequal
But the √ sq.root. that I don't understand.
The legs “waking toward” the answer actually reminded me of greater than/less than. When I was a kid, the way I came up to remember which was which was that the symbol was a mouth “talking to” the bigger number.
Actually, I come from Belgium and we use the dot a lot for multiplication especially in secondary education. Also, it's doesn't really matter if you can't see the dot because multiplication can most of the time not be written at all 😄
Particularly in algebra where 2y = 2 × y = y + y.
In Germany we use multiplication ⋅
For division we use : or ÷
+-=% are the same
I've seen my Polish parents use " • " instead of the " × " like in my British school.
I'm American I was taught the same thing in 5th grade, but before 5th grade I was taught to use ×
I literally searched this on google and the next thing you know, this video pops up in my feed.
Percent is % - there is also the "permille" which is out of 1000, and looks like ‰
More people see "ppm" and "ppb" than "per mille" and "per myriad".
"Whoever thought it was a good idea to put Letter's in Math. I just want to talk to him."
Fitting meme I'd say.
in my schooling, the × and ÷ symbols were only really used in elementary school. Even though i rarely do math these days if i ever write down an equation , 100% of the time i'm using the dot for multiplication and a fraction notation for division. it makes a million times more sense and infact the multiplication and division signs introduce a level of ambiguity that is completely unsuited for doing math
Very interesting! Never thought about this although I’m interested in etymologie
In Finnish school (I started around mid 2000s) we never used ×. Not even any teacher had it in personal use. Because of this I've always seen it as a foreign thing limited to some calculator apps.
Another awesome video! thanks!
Will there be a sequel to this topic covering other math symbols such as the Square Root, less than ()? Also, because I heard you say "a bit of a faff," could you explain British phrases that sound odd to non-Brits, such as gutted, chuffed, and wind-up
Why are the dots not central, but slightly to the left? (Relative to the stroke) 6:13
interesting that this etymology of the percent symbol has a letter 'o' underneath, where the per mill symbol (‰) must've been based on the lower number looking like a 0.
the delta symbol (Δ) if often used to represent the change in value between two quantities
Turn it upside down and you get a symbol used in vector and scalar fields.
My personal theory about the % symbol (which I long assumed would never be more than a theory) is rooted in its meaning.
Since "percent" literally means "of one hundred", it was probably originally notated as "/100". Of course, two near-vertical lines written rapidly in succession would cause legibility issues, the 1 was eventually dropped to form "/00". Similar to the evolution of the ampersand (&), this gradually morphed into "0/0"; making the relationship with % pretty darned obvious.
You ever want your brain to hurt, read an ancient Greek mathematics/geometry text as it originally appeared: all words, no numbers or math symbols.
Except where there is a symbol in some diagram with absolutely no explanation as to what it means.
We use : for division in Bulgaria and mostly . for multiplication. For me, percent % and promile ‰ are just simplified versions of 1/100 and 1/1000
Tell me how sign for pro mille came to be - ‰ and how it is connected with %, if % is shortened pro centum.
The × and • symbols actually have slightly different meanings in physics. It's hard to explain fully (perhaps someone can help?) But it has to do with combining tensors. Essentially, the × or "cross product" is just the raw multiplication of all of the numbers, while the • or "dot product" takes into account all of the magnitudes and directions of all the different tensor values in relation to one another to determine a net vector
The dot for multiplication would cause confusion as it could be confused with the decimal point (in the English speaking word) or the thousands seperator as used in Europe.
Ways if saying + -
Plus and Minus - common language
Add(ition) and Subtract(ion) - more "professional" or formal
Positive and Negative - the value itself
in computing, % is used as the modulo (often shortend to just "mod") operator symbol i.e. the remainder after integer division e.g. 11 % 2 = 1. There isn't a single symbol (that I am aware of, at least) to represent integer division, but in the programming languages I know best, it is "//" e.g. 11 // 2 = 5. It's just the counterpart of modulo
Why do mathematicians use the abbreviation "mod" if "%" is more concise?
@@Anonymous-df8it probably because it would look like percent in that context
@@BrennenRaimer Why would that be an issue in mathematics but not programming?
@@Anonymous-df8it because computers are deterministic and do not have the possibility of misinterpreting the input. To the computer, % is unambiguously the modulo operator and percentage is not an operation unto itself. To take the percent of something you would have to do a multiplication and division operation (* and / respectively)
@@BrennenRaimer But no-one uses percentages in professional mathematics!
What about set notation?
I see the divide by symbol as a dash dividing a colon into its two component dots.
Me subscribed + me up late = me early
Over here in Lithuania, we always use the dot! At least when writing by hand.
I have always thought that the division symbol is a line dividing two dots. Two divided in two, is one on either side. That's probably wrong, but I thought of in second grade (age 6)
I always imagined that the % sign was made by just taking 100 and putting the 1 between the 0's: 010 -> %
In computer languages, the double asterisk (**) or the upwards pointing arrow (↑) are used to indicate exponentiation (raising to a power). The older computers simply had no way of superscripting numbers the way that this was traditionally done. Frankly, it’s awkward even on modern computers and is very difficult to see.
The latter symbol is used in Knuth's up-arrow notation and is instrumental in the calculation of Graham's Number. Nowadays we use the carat (^) symbol in lieu of the up arrow for typed exponentiation.
Using the dot to mean "multplied by" is a bit confusing. It means you have to ask, 'Does 4•7 mean "four times seven" or "four point seven" (i.e. four and seven tenths)?'
i've never really seen it used for multiplication of numbers, tbf, only in algebra
RIP for anyone using earphones during the transition to the last bit
i disagree with one of your statements.
5:28 i think cdot (center dot) is actually closer to perfect (as long as you have good pen and/or vision or attention), as someone that use nothing or cdot as multiplication. i only use × for cross product of vectors.
i would argue we should ditch × as multiplication sign at all education levels, but cdot can go unnoticed as you said in video, so i think every teacher should teach multiplication sign as tiny cross that is centered on the grid/line, and encourage kids to make the cross tinier each year (or it might be how it is being taught in some places rn, idk)
My brain: “oh yeah, cent! Like centaur!”
Centaur (according to google) actually comes from Kentauros, the name of a tribe of expert horsemen
This video reminds me of a college prof who insisted we refer to the sign of a number (like -23) as siggen, so it wouldn't be confused with sine.
*signum
I was taught that the % is just a one and two zeros and thats why per mille, which is per thousand is a one with three zeros ‰
same
5:26 Wait til this guy finds out we barely use any multiplication symbol when multiplying scalars anymore
Or division symbol for that matter...
1:03 Then there's me who watched this right after watching a video on the three cube roots of -1...
Are you referring to -1, 1+sqrt(-3)/2, and 1-sqrt(-3)/2?
@@Anonymous-df8it Close. -1, (1+i(sqrt(3)))/2, and (1+i(sqrt(3)))/2. The /2 needs to apply to both the 1 and the sqrt(-3)(or i(sqrt(3)))
@@TheJaguar1983 Oops! Should I correct my previous reply? Also, should I delete this one? Please answer both individually
@Anonymous-df8it Up to you. I personally prefer to leave records as is, but sure, clean up the second question once you decide.
Math was my favorite subject as a kid. and Throughout school! ^^
in the divide sign the 2 dots have been divided by the line
The division symbol looks like a group of two dots being divided in half. Maybe that's the origin?
I saw somewhere that the percent sign % is a fraction that made its way into writing. So, 10/100 being ten out of 100 or 10%
Has Name explain made a video on Monarchical Ranks or Knight Ranks?
isnt the approximately equal sign two tildes?
saying that letters don't play much role and that the multiplication dot is unpopular is very alien to me. "Math with letters" was a primary school thing for me, and when you look at math that has anything to do with the real world, it's full of letters. Keeping the multiplication cross after primary school math is just not going to work. Multiplication is also extremely common in applied math, so a more condensed notation is strongly favored. When possible, multiplication is indicated by just putting the factors next to each other, but when that's impractical or ambiguous, it's *always* the dot.
Cross products exist?!
Maybe the division symbol is two people and a wall in between them?
Wouldn't the resemblance between the multiplication sign and the letter x depend upon the font or handwriting style?🤨
Yep. They're pretty easy to distinguish in any font meant for math, but you're generally taught that the multiplication sign is just a floating x so it's hard with most people's handwriting. But I'm definitely also not the only mathematician who's deliberately changed my handwriting to make my math notes easier to read...
One of the problems with saying "maths" instead of "math" is made obvious with this video. Trying saying "maths symbols" three times quickly. :)
Kind of surprised you decided not to mention two other quite popular means for annotating multiplication -- the asterisk (*) and of course, just simple parentheses between neighboring terms (both of which are almost always more popular than the classic × in typed rather than handwritten math)
Also, a percent is not "normally out of 100," as you said; that's the literal _definition_ of "percent." It's *always* with respect to 100.
Me vs Myself & You vs Yourself? Not valuable distinctions? You are teaching people!
There's also a version of equals with three parallel lines, which means "Exactly Equal To".
What you refer to is the triple bar used mostly in logical mathematics to denote that x is logically equivalent to y and outside of that context it means x is identical to y.
It can also mean congruence. Two expressions are congruent if they equal each other no matter what the variables are set to. Two numbers are congruent modulo a third if they yield the same remainder after dividing by the third
the division symbol low key looks like a numerator over denominator
like x/y == */* symbol
Please don't use & instead of +. It is already in use in programming as a completely different operation, combining the binary representation of numbers, making a bit 1, only if it is 1 in one of the numbers. So 2 & 3 = 2 (0010 & 0011 = 0010).
So, binary addition without carries?
The word 'mathematics' is plural for a reason. Its abbreviation should end in 's'.
But "mathematics" is not usually used as plural in English. It's most often used as a singular noun in modern usage. (e.g., "Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes." - en.wikipedia.org)
French does still usually treat it as a plural noun: « Les mathématiques sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations, etc. » - fr.wikipedia.org
I mean, 'mathematics' behaves grammatically in the singular (since it's uncountable; one does not go "one mathematic, two mathematics..."). Also, my computer hates it if I spell it like that...
...I'd agree with you on the spelling however in writing or in speech, but I'd rather just not abbreviate it when typed, since I don't like the computer complaining
"The M was removed entirely"
Didn't the percent sign come from the number 100 010 ⁰/0
It also is the reason why the permille sign has 2 zeroes at the bottom ‰