Does omitting the multiplication operator have an effect on order of operations

Question

When you write a mathematical expression like this:

$4:2(1+1)$,

does the fact that the multiplication operator is not explicitly written has any bearing on the precedence? What is the order of operations in this case?

Is it:

$4:4=1$ (order: parenthesized addition, implicit multiplication, division)

or

$2(1+1)=4$ (order division, parenthesized addition, implicit multiplication).

If the explicit operator has no effect, this would be $4\div2\cdot(1+1)$ and calculated from left to right (because of the no precedence between division and multiplication). Result woud then be $4$.

Answer 1

Implicit multiplication belongs to algebra, not arithmetic. I would not expect to see implicit multiplication in an expression like yours - where there are unevaluated operations involving literal numbers. This could clearly lead to confusion, as we might hope to simplify $4(2+2)$ to $4 4$, but this is indistinguishable from the number $44$.

Where implicit multiplication is appropriate, in algebraic expressions, between a number and a symbol, or between two symbols, neither $:$ not $\div$ should be used to express division. Rather division should be shown using a horizontal line. This means there is no confusion possible between

$$\frac{a}{b}\left(c+d\right)$$

and

$$\frac{a}{b(c+d)}$$

Answer 2

Let me answer by giving a stupid example: $$2a:2a$$ Just adding the implicit multiplication sign will lead to this result, by calculating from left to right: $$2*a:2*a=a*a=a²$$ To me this is ridiculous, I think it's quite clear that the answer should be 1, because I think writing factors together without multiplication sign "binds them" tighter together than explicit multiplication or division signs do. I read it like this: $$(2a):(2a)=1$$

So my answer to your question is: Omitting the multiplication sign shouldn't change the evaluation of mathematical expressions, but sometimes it might. However, this confusion is completely unnecessary. Mathematics is a language, and expressions like the one I wrote does not exist in themselves, they are a product of human communication, and writing the expression with a fraction line would have eliminated all ambiguity, as pointed out by jwg.

If one means 2a divided by 2 and multiplied by a, one shoud write $$\frac{2a}{2}*a$$ If one means 2a divided by 2a, one should write $$\frac{2a}{2a}$$ If you are tasked with solving an expression like the one you posted or the one I mentioned, from your teacher or whoever, you could critisize them for poor mathematical communication.

Answer 3

The right way to do it, I think, would be as follows: $$ 4 \div 2 (1 + 1) = 4 \div 2 \times (2) = 4 \div 2 \times 2 = (4 \div 2) \times 2 = 2 \times 2 = 4. $$

However, one can also proceed as follows: $$ 4 \div 2(1+1) = (4 \div 2) \times (1+1) = 2 \times (1+1) = 2 \times 2 = 4. $$

Note that parentheses carry the topmost precedence.

Except for the grouping operators such as the parentheses, the multiplication --- whether expressed with or without the sign --- and division have equal precedence, followed by addition and subtraction, which also have the same precedence. However, operators (of the same precedence) are to be evaluated from left to right.

Hope this helps.

Answer 4

When you write a mathematical expression like this:

4:2(1+1),

does the fact that the multiplication operator is not explicitly written has any bearing on the precedence? no What is the order of operations in this case?

If the : means division, which i assume it does, then the 4 is divided by the expression 2(1+1). It is a fraction. Then you are free to divide the 4 in the numerator by the 2, then divide by the sum of 1+1, or multiply by 2 thru the denominator (and get 2+2), then add, and then divide, or add 1+1 first before the multiply thru by 2. In any case you get 2/2, or 4/4, =1.

Answer 5

Technically the implied multiplication is really just hiding the real controversy here.

By foil, yes, of course the term outside the brackets must be able to be distributed inside of it, or all of Math breaks down.

The some people have nonsensically argued that the equation is better parsed as: $$\frac{4}{2}(1+1)$$

But you have to be aware that smart calculators might choose either interpretation and teachers might choose 1 of these randomly and consider it correct. Their are even some stories of math books that teach implied multiplication but the person who wrote the answer section did not use it. So it is indeterminate, nobody important enough has given a ruling on this and enough people have chosen both sides that their is no way of knowing the implied meaning.

Answer 6

There's a number of answers here which are kinda correct, but have missed using the correct terminology and reasons, and the correct term is, co-incidentally, :-) Term...

Terms are separated by operators and joined by grouping symbols. In terms of the "absent" multiply, that means it's now 1 term instead of 2. i.e. ab=(axb), not axb. ab is 1 term, axb is 2 terms.

If a=2 and b=3, then...

axb=2x3

ab=6

1/axb=1/2x3=3/2

1/ab=1/6

So yes, it makes a world of difference.

Note that in the mnemonics, "Multiplication" refers LITERALLY to multiplication symbols, nothing else. axb is multiplication, ab is a term. In fact it's a product, which is the RESULT of a multiplication. In other words, it's already been done. e.g. ab=6. This is how an awful lot of people get order of operations questions wrong - they think that ab is "multiplication", and thus break up the term, giving the wrong answer. e.g. doing 1/ab as though it's 1/axb, when it's 1/(axb).

Also, in the specific example you've given, The Distributive Law also applies. i.e. a(b+c)=(ab+ac). In contrast, ax(b+c) is 2 terms, not one, so 1/a(b+c)=1/(ab+ac), whereas 1/ax(b+c)=(b+c)/a, which is quite clearly not the same answer.

P.S. this is WHY it's NOT called "implicit multiplication" - it's not multiplication! No Maths textbook says "implicit multiplication" - they all talk about Terms/products. "implicit multiplication" is a "rule" made up by people who don't remember the real rules - Terms and The Distributive Law.