Is there any specific convention to how mathematical equations are "read out"?

Question

Often, mathematical information and formulas are shared through text, maybe on books or on websites. If you see the equation:

$$x + 3 = 4y$$

we can easily read it out as "x plus three equals four y". But for some other equations such as:

$$\frac{x+3}{x-7}=y$$

it doesn't seem too obvious on what it should be read out as. I would read it as "x plus three the whole divided by x minus seven equals y", however someone else may read it out differently.

My question is therefore: If in a situation where oral communication of mathematical equations is necessary, does there exist a convention to read out mathematical equations such that one particular equation is read out in only one way, and that it is possible to deduce the equation from the spoken name alone?

Answer 1

I would like to elaborate on Hagen von Eitzen's insightful comment:

The different groupings / implied parantheses are often expressed via pauses and prosody.

I think this is exactly right and plays an extremely important role in disambiguating oral mathematics. There is a huge difference between:

"X plus 3, over X minus 7, equals Y"

and

"X, plus 3 over X, minus 7 equals Y"

and

"X plus 3, over X, minus 7 equals Y"

If the cadences and pauses in the sentences are read aloud consistently, I think it is unlikely that anyone would transcribe one of those in place of another.

However, in some cases (particularly for extremely complicated expressions with groupings nested inside other groupings) one may call attention to the grouping using verbal annotations like "the quantity" or "all". For example, the quadratic formula is often read aloud as

• "X equals negative b, plus or minus the square root of the quantity b-squared minus 4ac, all over 2a".

Answer 2

I would read $$\frac{x+3}{x-7}=y$$ as:

Quantity $x$ plus $3$ over quantity $x$ minus $7$ equals $y$.

Answer 3

with computing there is something called reverse polish syntax, where operators follow operands, this allows all expressions to be done without brackets!

eg $3+4$ becomes $3,4,+$,

$(3+4)*(5-6)+7$ becomes $3,4,+,5,6,-,*,7,+$

so $\frac{x+3}{x-7}=y$ becomes $x,3,+,x,7,-,y,=$

The general format is

operand1 operand2 ... operandm operator

its probably not practical for maths because it is obfuscating, but it is used in some computer languages BECAUSE its much easier to implement than normal maths. eg if you want to implement a computer language in a hurry you will opt for reverse polish as it gives maximum expressive power for minimum implementation work.

Although Tex depicts things 2 dimensionally, you could linearise that via brackets, namely $(x+3)/(x-7)=y$ and then read out the brackets eg: left bracket $x$ plus $3$ right bracket over left bracket $x$ minus $7$ right bracket equals $y$

I would only say out things without mentioning brackets where the expression is not too complicated, and I would say eg

xplus3 over xminus7 equals y

where I say without pause xplus3 to indicate its a group.

If everything so far is the upper part of the fraction eg

$\frac{\frac{x+3}{x-7}+9}{x+10}+4=y$

I would say "all over", ie everything so far over the next thing, where a lack of pause on what follows means the lower part of the fraction

so the above becomes:

xplus3 over xminus7 plus 9 all over xplus10 plus 4 equals y

ultimately you need a protocol, and its a question of grammar, where you have precedences and associativity. With grammars the big problem is ambiguity, so usually you need disambiguating rules, eg left associative or right associative, and precedences.

eg $x-y-z$ is mathematically ambiguous because it could be $(x-y)-z$ or it could be $x-(y-z)$ which are NOT THE SAME! The problem is which $-$ do we apply first? The disambiguation is to make $-$ left associative which means we apply the leftmost one first. This left associativity is a hidden rule of maths which most mathematicians arent aware of except intuitively, but in computing this nettle has to be grasped by someone.

with $+$ and $*$ it doesnt matter which we do first because these 2 operators are associative, but $-$ and $/$ are not associative. eg $2/(4/2)=1$ but $(2/4)/2=1/4$.

But although $+$ and $*$ are associative, there is a technical matter with computing that $x+(y+z)$ ISNT always the same as $(x+y)+z$ because of the problem of overflow, eg if $x=$MAXIMUM_NUMBER, $y=1$, $z=-1$, then $(x+y)+z$ will lead to overflow, whereas no overflow for $x+(y+z)$. Also you can run into problems with precision, where the 2 expressions can give slightly different answers eg because there are say only 8 digits of precision. eg $((-1.2345678) + 1.2345678+0.00000001$ gives $0$ if you do the right $+$ first, but gives $0.00000001$ if you do the left one first.

Numerical analysis is the subject which looks at overflow and optimising precision and other computer specific maths things.

mathematicians often run into problems with computers because computer maths is not quite the same as mathematicians' maths. At school I wrote a program to simulate gravity with a planet orbitting a star, but the planet spiralled into the star because of rounding errors!

another example $x+y*z$ this could be $(x+y)*z$ or $x+(y*z)$ in fact its the latter, where in fact we do the right operator first BECAUSE $*$ has higher precedence than $+$.

so I would personally say the best protocol is to linearise with brackets and then just read out the symbols as that minimises the grammar to the level of the symbol description.

so the last expression linearises to $((x+3)/(x-7)+9)/(x+10)+4=y$

and you just read that out symbol by symbol, and you could optimise the protocol by just saying "left" for "(" and "right" for ")".

namely:

left left $x$ plus $3$ right over left $x$ minus $7$ right plus $9$ right over left $x$ plus $10$ right plus $4$ equals $y$

You can evade the problems of associativity and precedence via total bracketing.

basically as someone who has dabbled with computing and maths, with maths, people tend to work with a fixed protocol, whereas with computing people will dabble with the protocol eg the grammar to get around constraints. So eg with pictures, one protocol is jpegs, another is pngs another is tifs. Its the same data just via different protocols.

So I would agree with the other people involved an optimal protocol for the communication channel being used. eg LateX is using a visually optimal protocol which you cant use in an ordinary text message eg an ordinary email.

so in an ascii .txt file, I will write Integral for the integral symbol as the channel constrains my symbols to the ordinary symbols.

LateX itself is a linearised format for the 2 dimensional formatting!

so in theory you could communicate the visual format by reading out the linearised LateX! Furthermore LateX is based above Postscript and Postscript uses a form of reverse polish syntax.

eg $8 − (7 × 3)$ in Postscript is $8$ $7$ $3$ mul sub

or equivalently as $7$ $3$ mul $8$ exch sub