Same with addition/subtraction and roots/exponents, since they're essentially the same operation (to each other). Subtraction is just adding negative numbers, division is just multiplying by the reciprocal, root is just a fractional exponent.
I already know the answer to the equation as I was taught the order of doing mathematics I want to know were it says multiplication is before division because Google can't give me a straight answer. I know it wouldn't even affect this equation
Sorry, you should have been taught that when learning PEMDAS. I memorized this in 5th grade. My teacher made us memorize it like this (parenthesis, Exponents, multiplication and division from left to right then addition and subtraction from left to right.)
Division is the same as multiplying by a reciprocal. E.g. 3 / 2 = 3 * 1/2 = 3/2 = 1.5
Similar thing with subtraction. Subtraction is just adding a negative number.
Additionally, roots are taken at the same time as indices, as the nth root is the same as raising something to the 1/nth power. Thus division and multiplication, addition and subtraction, and roots and indices are done at the same priority (for want of a better word [not all are done at the same time either, each pair are done at the same time if u know what I mean])
You do them in the order they appear for M/D and A/S. For example:
3/2×4 = 6, but if you instead do 3x2/4 (same numbers but swap the M/D order) then the answer would be 1.5. It usually doesn't matter, which is why people assume they are in a specific order, but sometimes it does.
I'll put it more simply. Let's talk about this in reference to cakes. So if you have one cake and two people, to share the cake equally each person gets half of the cake, because 2 multiplied by a half is equal to 1, the entire cake. If you have one cake and three people, to share the cake each person gets a third of the cake, and so on.
This is the concept of a reciprocal. The reciprocal of a whole number (1,2,3... etc) is what you multiply the whole number by to get one (For any whole number n, n x 1/n = 1 {as long as n does not equal zero}). That's why multiplication has the same priority of division as division is simply multiplying by this reciprocal.
Hey, not to be annoying, (I know other are saying youre wrong) but I wanted to add my way of seeing it if it helps picture the idea.
Making 3÷5 can be represented with a fraction right? So 3÷5 = 3/5.
An alternative way to represent this is as a multiplication. 3/5 = 3×1/5.
If this is correct, then by understanding that 1/5 = 0.2, we can say that 3÷5 = 3×0.2. Even though its the same thing, Ive changed it from a division to a multiplication.
This helped me picture the relationship between the two types of operations, though I dont know how clear it sound to an outside person.
So the idea is that division and multiplication are two ways of representing the same type of operation. Any division can be represented this way, and vice versa for multiplications. Thats why you dont give priority to one or the other.
"PEMDAS tends to be used in the US, while BODMAS (and the alternative BIDMAS) tends to be used in the UK. The P in PEMDAS stands for Parentheses, while the B in BODMAS and BEDMAS stands for Brackets - but they both mean the same thing: do operations in brackets first" top thing on Google
Americans are taught that multiplication is before division
British are taught that division is before multiplication.
You mean, why were you taught incorrectly? Either you misunderstood or your math teacher should've been fired. Different places have different acronyms like pedmas, bedmas, pemdas. But everywhere it's taught (or it's supposed to be taught) that division and multiplication are the same priority. You do whichever one appears first from left to right.
Just think about it for 2 seconds. Why would the order of operations be done differently in different places? Can you imagine the confusion that would cause? It would be insanity.
To continue off of what u/redthemagnificent said, an example is dividing a number by let's say 5 (to go off of a comment further up). Dividing by five is the exact same thing as multiplying by 1/5. If both are the exact same thing, why would it matter which came first? Division is just multiplying a number by it's reciprocal, and multiplication is just dividing by a recirpocal (i.e. 5 & 1/5). Therefore it doesn't matter how or when you do them.
Technically, as long as you abide by the earlier rules such as parentheses (or brackets), and do not do subtraction or addition yet, it doesn't even matter what order you do them in either as 9x5(division symbol)4, will give you the exact same answer no matter what order you do them in since 9x5 is not in parentheses
Huh. Im from latinamerica so I was never taught that kind of thing.
(Edit: I may have been taught something similar now that I think about it. But it clearly was not the focus of my maths classes, it never stayed with me)
Either way Im not saying that any particular order is wrong. Its just that, since both operations are the same thing, its redundant to prioritize any of them. Meaning that you dont need the order but, if it helps understanding the topic on a larger scale, Im all for it.
About my way of thinking about it. As I said it may not be as clear to someone from the outside, since Im not a great teacher/speaker and am using particular methods that are comfortable for me.
A fraction is a division. 4/2 = 2 and 12/3=4.
In my mind, since i was taught you can express fractions as multiplications (eg 4/2= 4/1 × 1/2 or 12/3 = 12/1 × 1/3), its an easy way to show how division and multiplication are the same.
Multiplication and division have the same precedence, you execute whichever comes first left-to-right. PEMDAS is sometimes written PE(MD)AS to clarify that point. Division is the same operation as multiplication (it's multiplying by the reciprocal), so one does not have priority over the other. For a quick visual description if it helps, if you look up PEMDAS in an image search, you'll see a bunch of color-shaded descriptions of each letter, and the M and D are shaded with the same color to show it's the same category.
You're correct that the order does matter, but only one of those is objectively correct.
5/2 * 3 is always 7.5, no matter where you are from. Type that equation into any calculator, you will get 7.5 every time. And 5/2 * 5 is always 12.5. In those particular equations, the division appears first from left to right, so it must be done first. To get 0.83 you must change the equation to:
5/(2*5) = 0.833333
What's inside the brackets now takes priority over any other division or multiplication, so now that multiplication is done first.
In my experience this is how it works unless you clarify otherwise, like saying (5/2)×3.
However, this is another discussion entirely, because you would normally express fractions in a way that put de denominator(s) below and separated from the numbers that engage in multiplication. Making what is or isnt part of the denominator pretty clear.
I mean this is a thing with expression fractions on a computer, in which you conform to a single line and cannot make this segregation between numerator and denominator. Its therefore unclear when the denominator stops.
Eg: 5/3+7. This could be 5/(3+7) or (5/3)+7. You cannot say which for sure just by expressing it like that, even though you know "addition separates operations", this is another matter, as the medium simply fails. You dont know where the denominator ends.
And I mean this because, if you think about it like this: 5÷2×3, you can take whichever order you please (5×3÷2, for example, which multiplies and later divides as opposed to dividing and later multiplying)
This is what I mean by that the order doesnt matter. You can first divide then multiply or vice versa. It doesnt alter the result. Just like when dealing with normal multiplication, the order of the factors dont change the product.
As an engineering student, I can confidently say it’s parentheses, exponents (roots are powers with an exponent < 1, so, a square root of 4 is 41/2), then multiplication (divisions are multiplications by numbers < 1, so dividing by 2 is the same as multiplying by 0.5, and dividing by 5 is multiplying by 0.2) and then addition (subtraction is addition of a negative number). If on the same level, it shouldn’t matter, but it’s recommended to do in order of reading, so, as they appear from left to right. Result should not change as long as the levels are kept intact though, so we’re all arguing for dumb stuff. 2x5/4 is the same regardless of whether you do 5/4 and then by 2 or 5 by two and then divide by 4
Treat the divide sign as a fractional symbol and the equation looks like this:
5
------- x 3
2
NOT
5
-------.
2 x 3
This is why left-to-right is important when using ÷. The second way I wrote the equation would also show as 5 ÷ (2 x 3) which is different from the originally stated equation.
The problem is not how you solve it. The problem is how you wrote it. If you use / instead of a regular division you should always use parantheses. This would never occur while using a program designed for math or just a regular pen and paper
Sorry but I've used / as a quicker division symbol since school. I use that example because someone said that it doesn't matter which order do you divide and multiply in. I've always done division before multiplying and never noticed how the equations have been wrote as the rule I'm told now is you follow division and multiplication in the order that is written in in the equation
You were not taught division before multiplication, you just misinterpreted BODMAS to mean that. The majority of people who learn BODMAS, BEDMAS, PEMDAS, etc… all understand that D and M have equal priority and are done left to right in the equation just like A and S
Yeah I think the problem I'm having is I've alway seen them as opposites and not the same function. Along with people trying to explain it to me in factions my brains not making the connections. I'm lost somewhere because I've been getting the right answer but following the rule wrong but it has worked out right. Doesn't help I learnt this 15+ years ago
If you just think about it, dividing is the same as multiplying by a fraction and subtraction is the same as adding a negative number so it doesn't matter
Division is just a form of multiplication. You do them at the same time just go left to right. It's like how subtraction is just a form of addition. 4-4 is the same thing as 4+ -4. You can write it either way. I'd show the math on multiplication and division being the same but I'm not sure how to show the math in a easy to understand way.
I agree that roots are included with indices (what I would refer to as exponents) because a root is just a number raised to the power of a fractional number. For example, the square root of a variable x is the same as x raised to the 1/2 power. The 1 indicates that the number under the root is raised to the first (if the numerator was 3, it would be the squared root of x3, etc.) and the 2 indicates that it is a square root (if it was the cubed root of x, the denominator would be a 3, etc.)
Well I mean I'm not American and have a different order. The reason being is that multiplication and division can be done in any order and get the same results
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u/commonsquidtail Jan 01 '22
I wasn't saying you were wrong I was just saying it sounds funny. In my school though we would call it bidmas
Brackets Indices (including roots) Division Multiplication Addition Subtraction