Fundamentals of Math - old

Fundamentals of Mathematics

They cannot both be correct, so which way do you solve the problem? Well, mathematicians (as early back as the 1800s acknowledged this problem years ago and collectively began testing and organizing which operators took precedence over another. There have been some modifications over the years, but the general principles remain the same. The order of operations state: 1. Do everything inside the parenthesis first. 2. Then take care of any exponents or radicals left to right (whichever comes first). 3. Next, you should multiply and divide left to right (whichever comes first). 4. And finally, add and subtract - again left to right (whichever comes first). So how do you answer "what is 5 + 3 × 4 "? Multiply 3 and 4 before adding 5; making the correct answer 19. It's also important to note that operators can be displayed very differently from problem to problem.

Figure 1.5.1 Order of Operations

P

E

M

D

A

S

P lease

E xcuse

M y

D ear

A unt

S ally

Mnemonic

Parenthesis Exponents Multiplication Division

Addition

Subtraction

Parenthesis

(anything in here) or [here] or {here}

Exponents/ Radicals

or under here √

anything up here

Multiplication /Division

9 3

× ∙ ∗ 2(3) 8 ÷ 2

7 |21̅̅̅̅

Addition/ Subtraction

+ − ±

You will often notice too that when working lengthier problems, your expressions start to resemble an inverted triangle as you find your solution.

Example 1.5.1

1. There are no parenthesis or exponents (or radicals), so begin with multiplication and division. Division occurs first (left to right), so start by diving 30 by 5 (which is 6). 2. With the simplified expression you still need to take care of the multiplication before you add, so 6 times 2 is 12.

30 ÷ 5 × 2 + 1

6 × 2 + 1

3. Now add the remaining to values.

12 + 1

The solution is 13. 13 Now let's consider the case when you have multiple sets of parentheses (or nested brackets). In this instance, you should begin with the innermost set of parentheses and work your way out. Once all the operations have been completed inside the parenthesis, you can drop the parenthesis (or brackets) altogether.

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