DESDE LA ECONOMÍA

Negative ÷ POSITIVE = Negative Negative ÷ Negative = POSITIVE

The rules for dividing signed numbers THE SQUARE OF A NEGATIVE NUMBER

A common point of confusion among students is the difference between the square of a negative number, and the negative of a number squared. They are not inter-changeable. See the next example. We have -7 in paren-theses with an exponent of 2 outside. Remember that according to PEMDAS, we always must first evaluate what is in parentheses. In this case, there is nothing to do within parentheses. We then move onto the exponent which affects the contents of the parentheses. We have (-7) × (-7) or 49.

(−7)

= −7 × (−7) = 49

Squaring a negative number

In the next example we don’t have parentheses. Should we do our squaring first, and then make the answer negative, or do we square a negative number?

W O R K I N G W I T H N E G A T I V E N U M B E R S

−7

= −(7

) = −49

Making a squared number negative

The answer to that question involves knowing what it means to make a number negative. When we put a negative sign in front of a number, what we’re really indicating is to multiply the number times -1. We’ll work with that concept more later.

For the purposes of our problem, we have an exponent, and we have implied multiplication. According to PEMDAS, the exponent has higher priority. We first square the 7 to get 49, and then we make it negative with implied multiplication by -1, giving us an answer of -49.

THE SQUARE ROOT OF A NEGATIVE NUMBER

Until much later math, we say that the square root of a negative number is ‚undefined‛—it ‚cannot be done.‛

Remember that square root asks, ‚What number can we square in order to get this number?‛ Think about −16.

What number can we square in order to get -16?

Recall that squaring means to multiply a number times itself. When we do that we will always get a positive answer since the signs of the numbers match. We can’t end up with a negative number.

−16 = 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

The square root of a negative number is undefined

POSITIVE NUMBERS HAVE TWO SQUARE ROOTS

Now that we’re learning about negative numbers, let’s take the concept of square root a step further. Remember again that square root asks the question, ‚What number can we square in order to get this number?‛ Think about 16. What number can we square in order to get 16?

We’ve already seen that 4 is certainly an answer. If we square 4, we get 16. That means that 16 = 4. Are there any other solutions? It is wrong to say 8 × 2. Certainly those two numbers do multiply to 16, but they are not the same number. We need a number that we can mul-tiply times itself in order to get 16.

Let’s extend our search to the world of negative num-bers. Remember that if we multiply a negative number times itself, we get a positive answer. That means that if 4 was an answer, it stands to reason that -4 would also be an answer. (−4)² = 16, so that means that 16 = -4.

We say that the square root of 16 equals ‚plus or minus 4.‛ There is a positive and a negative answer, and both are valid. This is written symbolically as 𝟏𝟔 = ±𝟒.

W O R K I N G W I T H N E G A T I V E N U M B E R S

Even though there are two answers, we refer to the positive answer as the principal square root. This is the

‚default‛ answer we give. One reason is that square roots play a big role in geometry which you’ll learn about later. Geometry deals with distance which is always positive. In such practical cases we ‚discard‛ the negative evaluation of a square root, even though it is mathematically valid.

16 = ±4

The square root of a positive number has “plus and minus”

versions. The positive one is considered the “principal” root.

MORE ABOUT ABSOLUTE VALUE

We are sometimes asked to compute the absolute value of an expression, such as 4 – 6. That expression evaluates to -2, so its absolute value is 2. The absolute value of a number or an expression is always positive.

We use vertical bars around a number to represent the absolute value operation. For example, |-7| = 7. If there is an expression within the vertical bars, we must eva-luate that expression first, before we apply the absolute value operation to it. Review the previous sections on signed number arithmetic, then study these examples:

|3| = 3 |-7 + 6| = 1 |(-3) × (-5)| = 15

|-4| = 4 |10 – 8| = 2 | 20 ÷ (-4) | = 5

Computing the absolute value of various expressions

SO NOW WHAT?

Before progressing to the next chapter, it is absolutely essential that you fully understand all of the concepts in this one. In particular, if you do not fully master how to perform the four basic arithmetic operations with signed numbers, you will run into endless difficulty with all of the math that you will study from this point forward.

None of this is ‚busy‛ or ‚baby‛ work. It is the founda-tion of math.

Take as much time as you need to review the material in this chapter, and return to it as often as necessary until all of it becomes second nature to you, and you are no longer confused or intimidated by the sight of a negative number. See the Introduction and Chapter Twelve for information about how to get help, ask questions, or test your understanding.