Chapter 1, section 3

Order of Operations…using real numbers to compute

Regarding addition:

Addition is the first mathematical operation that you learned. It is important to see addition at a way of combining real numbers to get real numbers.

Addition with natural numbers is not usually a problem. Problems arise with integers and rational numbers.

When looking at a number that is an integer but is not a whole number you should see that number as –1 times a natural number.

For example, see –3 as a contraction of –1(3). This will help you later on when we start using variables regularly. For right now, we’ll use numbers.

We’ll also eliminate subtraction from our vocabulary and “see” only addition. This means adding a step to the problem 3 – 5…I want you to see it first as 3 + – 5 and then as 3 + – 1 ( 5 )…this sounds like work but it’ll pay off in the long run. I’ll be reminding you of this as we go.

In the absence of punctuation, numbers separated by “minus signs” are terms – numbers that are being added. So – 3– 5– 2 is – 3 + – 5 + – 2. If I meant for you to multiply, I am constrained to write the problem with parentheses: – 3 (– 5)( – 2).

Calculate– 7– 2

Now calculate

This is not the same problem as – 7(– 2 ) nor is it the same problem as . Let’s calculate these, too, just for practice.

Note that with addition, adding two negative number will result in a negative number; adding a negative number and a positive number results in a number that is the same sign as the number with the largest absolute value.

For example:

Regarding multiplication:

Multiplying numbers of the same sign results in a positive and multiplying numbers of differing signs results in a negative.

Let’s look at examples:

We will regard division as a special case of multiplication. Often I will remind you that division by a number or an expression is simply multiplication by the reciprocal of that number or expression.

What is the reciprocal of 3?

What is the reciprocal of ?

Using this way of looking at a division problem, let’s work:

So now is this a negative number or a positive number?

In Manglish, and reviewing inequality symbols,

______0

If you have more than two numbers, you multiply them pairwise using the rule for signs until you’re done.

Let’s look at this problem:

Here’s another problem:

Remember that dividing by number is the same as multiplication by it’s reciprocal. So this problem is

Let’s try a few more of these.

Regarding exponents and radicals:

The most important thing to remember is that the exponent only applies to the nearest number and not the whole expression.

Let’s look at some examples:

Putting it all together

When you have little or no punctuation, there is a definite order to the way you work problems. Work left to right on each step.

1st You work the insides of all parenthesis and other grouping symbols (fraction bars, radicals, braces, absolute value bars). You follow the steps below in order until you have performed all operations working from the innermost grouping symbols outward.

2ndyou perform all exponential and radical operations

3rdall multiplies and divides

4thall adds and subtracts

More problems