Forgive me if this is a stupid question but I can't find anywhere else to ask it. Suppose we had, $X + (X+5),$ is it correct to say we can "distribute" the plus sign into the bracket to get $X + X + (+5),$ to get the correct answer of $2X + 5,$ and as it works, is it purely by convention we don't write $+(+5)$ and instead chose to write $+5,$ thanks
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I see nothing wrong with this. – Bumblebee Oct 29 '22 at 11:39
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1You use the associative property to get $X+(X+5)=(X+X)+5=2X+5$. The distributive property concerns the multiplication of a number and a sum. You could say $1(X+5)=1(X)+1(5)=X+5$ but that doesn't seem to get us anywhere. – John Douma Oct 30 '22 at 14:30
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This began as a comment, but it got a bit out of hand (i.e. too long). Since I suspect you might not get a really basic answer, I went ahead and made this an answer. I tried looking for similar discussions in MSE about dealing with signed numbers, but I couldn't find anything sufficiently basic for what I think is needed here.
When I first studied negative numbers in school --- 7th grade, ages 12-13, this being 1971-1972 in a rural U.S. location; of course, I knew about negative numbers well before this, and I even knew about complex numbers at least a year earlier from library books such as those mentioned here --- positive and negative numbers were written as $^-2$ (= negative $2),$ $^+7$ (= positive $7),$ $^-1$ (= negative $1),$ etc. (I think we only dealt with integers for the first few weeks.) Every integer had an understood (left superscript) sign associated with it, except zero. We learned rules, described geometrically on a number line and described "algebraically", for adding and subtracting "signed" integers (and a short while later, rules for multiplying and dividing, which were actually simpler than the rules for addition and subtraction). For example, $^+7 \, + \, ^-3 \, = \, ^+4\;$ and $\;^-2 \, - \, ^-8 \, = \, ^+6\;$ were described geometrically on a number line and in an "algebraic" way (e.g. subtracting a negative number is the same as adding the corresponding positive number).
Thus, the signs $+$ and $-$ were used in two ways --- as part of the symbol used in writing the number (but when doing so it was as a left superscript; however, some textbooks may not employ this method of keeping the two uses of $+$ distinct) and as symbols used in denoting the operations of addition and subtraction. After a while we were weaned off the use of superscript signs for positive and negative numbers. This probably took a couple of months, and might not have happened at all that school year since I believe the introduction of negative numbers didn't take place until the last few months of that school year. (Math education then and where I was proceeded quite a bit more pedestrian than what it seems most everyone here has gone through.)
In your example we initially have $X + (X + \, ^+5).$ I'm not putting a superscript sign on $X$ because the symbol $X$ is intended to represent the number itself, regardless of how it is written. Using the associative law this equals $(X + X) \, + \, ^+5.$ Since $5$ is understood to mean $^+5,$ we can rewrite this as $(X + X) + 5,$ or $2X + 5.$ Alternatively, this can be considered the result of applying the "algebraic" rule for how one adds a positive number to something else.
The main point to all this is that technically you never actually have $+(+5).$ If $+$ means 'addition', then each of the two occurrences of $+$ is not correctly employed, since we need a number before $+$ and a number after $+.$ If $+$ is part of naming the number, then the left-most $+$ is not correctly used since the sign of a number is supposed to be to the immediate left of a numeral, not to the immediate left of a left parenthesis symbol. If the two $+$ symbols refer to different things (one being part of naming the number, the other being the operation of addition), then the symbols are still not correctly used (besides having the same symbol used for two different things).
Dave L. Renfro
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@Mathguy: the first plus means "add x and "+5" together --- I thought you were asking what $+(+5)$ means by itself in isolation (i.e. as a separate sequence of $4$ characters), and not what $+(+5)$ means as part of $X + X + (+5).$ As part of $X + X + (+5),$ the meaning of $+(+5)$ is (most likely) that in which the first $+$ represents the addition operation (a binary operation) and the second $+$ represents the sign of $5$ (a unary operation). – Dave L. Renfro Oct 29 '22 at 16:04
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However, one more thing that i have realised, say we had, x - (x+1), this simplifies to (x-x) -1, so its like 0-1, giving answer of -1, but most people would just cancel the two x's', going straight to the answer of -1. is my way the actual full method of why the answer is -1? sorry if this is a dumb question – Nav Bhatthal Oct 29 '22 at 16:44
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To be honest, questions such as how to fully/properly expand and rewrite something like $x - (x+1)$ depends on what the "arithmetical rules of engagement" are (i.e. axioms and arithmetical facts). One way is that subtraction is the same as adding the additive inverse, so $x - (x+1) = x + (-1)\cdot(x+1) = x + [(-1)\cdot x + (-1)\cdot 1 = [x + (-1)\cdot x] + (-1)\cdot 1 = [x + (-1)\cdot x] + ,^-1 = 0 + ,^-1 = ,^-1 = -1,$ where I've used various properties (distributive law, additive inverse is obtained by multiplying by $-1,$ and other things. (continued) – Dave L. Renfro Oct 29 '22 at 18:14
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Chapter 1 of Modern School Mathematics. Algebra 2 and Trigonometry by Mary P. Dolciani (1970) is one place that you might find useful to look through. I'll try to integrate some of what we've written in comments into my answer at a later time (especially my misunderstanding of what you meant by singling out $+(+5)$) at a later time, as I recently spent some time writing an answer to another MSE question and need to return to other stuff I have to take care of today. – Dave L. Renfro Oct 29 '22 at 18:15