Special cases (0 and 1, reciprocals, fraction line)

So it's true that you all know the basic order of operations -

1. Parentheses
2. Roots and Exponents
3. Multiplication and Division
4. Addition and Subtraction

But hey, there are some special cases that come between these steps that you should really know about!
Meet the special cases:
The number $0$, the number 1, reciprocal numbers and fraction line!

The number $0$

Addition operation

If $0$ appears after addition, it has no significance and should simply be omitted. With or without parentheses, if you add $0$ to any number, the number remains the same.

For example:
$(4+0)\cdot2+1=$
The addition operation with $0$ is unnecessary, the $4$ remains $4$.
Therefore we are left with the expression $4\cdot2+1=9$

If $0$ appears after a subtraction operation, it has no meaning and can be omitted. If we subtract $0$ - meaning nothing, from any number, it will remain the same number.

Pay attention -
If $0$ appears before the subtraction operation, meaning if we subtract any number from $0$, we get a negative number.
For example:
$0-7=-7$
The resulting number is the same number we subtracted, just in its negative form.

If $0$ appears next to a multiplication operation - regardless of whether it's on the right or left, the entire expression becomes zero.
For example:
$(5855\cdot9358)\cdot0=0$
In this case, for instance, there's no need to calculate everything inside the parentheses, and it's better to simply see that everything becomes zero because there's a $0$ next to the multiplication operation that zeroes out the result of the expression in parentheses.

If we divide $0$ by another number - meaning $0$ is on the left side of the division operation - the answer will always be $0$.
For example:
$(234+3434):0=0$
It is not possible to divide a number by $0$ and if such a thing appears you can write "undefined"
For example: $5:0$ = undefined

The number $1$

Addition Operation

In addition there is nothing new, we add $1$ to any number.

In subtraction there is nothing new, we subtract the digit $1$ from any number.

Multiplication by $1$ keeps the number you multiplied identical. In order of operations, this can help you solve exercises easily, for example:
$(233434:1)+2+1=$
$233434+2+1=233437$

In division, the number will remain the same if we divide it by $1$.
$21:1=21$
If we divide $1$ by a number, we get a decimal fraction.

The better you know the trick of reciprocal numbers, the more you can "skip the calculation" and continue the exercise easily!
Reciprocal numbers are two numbers whose product equals $1$.
For any number that is not $0$, the following is true:
$a\cdot\frac{1}{a}=1$

Examples of reciprocal numbers:
$2, \frac{1}{2}$
$54, \frac{1}{54}$
Remember – simply put $1$ as the numerator and the number as the denominator and you get a reciprocal number.

Dividing a number by any number is equivalent to multiplying that number by the reciprocal of the divisor.
That is:
$\frac{a}{\frac{1}{b}}=a\cdot b$
Note - The multiplication and division formula with reciprocal numbers can be very helpful for quick solutions following the order of operations.

You surely know that a fraction line is treated like a regular division operation!
It is known that:
$10:2$
is like
$10/2$

But! Remember to keep in mind when it comes to order of operations:

• If you have a fraction and there is any arithmetic operation in the numerator - you treat the numerator as if it is in parentheses. You handle it first before any other arithmetic operation.

For example:
$\frac{32-20}{6}+3\cdot2=$

Solution:
In this exercise, we see a fraction with subtraction in the numerator. According to what we learned, we immediately approach the subtraction operation as if it's in parentheses.
Let's solve the numerator and continue:
$\frac{12}{6}+3\cdot2=$
Now we continue with multiplication and division:
$2+6=$
And solve normally:
$2+6=8$