Problem 66 - Algebra and Trigonometry

Question

Signs of Numbers Let \(a, b,\) and \(c\) be real numbers such that \(a> b >0,\) and \(c < 0 .\) Find the sign of each expression. $$ \begin{array}{llll}{\text { (a) }-a} & {\text { (b) }-c} & {\text { (c) } b c} \\\ {\text { (d) } a-b} & {\text { (e) } c-a} & {\text { (f) } a-b c} \\\ {\text { (g) } c^{2}} & {\text { (h) }-a b c} & {} & {}\end{array} $$

Step-by-Step Solution

Answer

a: negative, b: positive, c: negative, d: positive, e: negative, f: positive, g: positive, h: positive.

Step 1: Analyzing Expression (a)

The expression is \(-a\). Since \(a > 0\), the negative of a positive number is negative. Thus, the sign of \(-a\) is negative.

Step 2: Analyzing Expression (b)

The expression is \(-c\). Given \(c<0\), \(-c\) turns the negative into a positive. Therefore, the sign of \(-c\) is positive.

Step 3: Analyzing Expression (c)

The expression is \(bc\). We are given that \(b > 0\) and \(c < 0\). The product of a positive number and a negative number is always negative. So, the sign of \(bc\) is negative.

Step 4: Analyzing Expression (d)

The expression is \(a-b\). Since \(a > b > 0\), the difference of two positive numbers where the first is larger is positive. Therefore, the sign of \(a-b\) is positive.

Step 5: Analyzing Expression (e)

The expression is \(c-a\). Here, \(c < 0\) and \(a > 0\). Subtracting a positive number from a negative number results in a negative number. Thus, the sign of \(c-a\) is negative.

Step 6: Analyzing Expression (f)

The expression is \(a-bc\). From Step 3, \(bc\) is negative, and \(a > 0\). Subtracting a negative is equivalent to adding a positive, making \(a - bc\) positive. Therefore, the sign of \(a-bc\) is positive.

Step 7: Analyzing Expression (g)

The expression is \(c^2\). The square of any real number is non-negative. Since \(c < 0\), \(c^2\) is positive. Therefore, the sign of \(c^2\) is positive.

Step 8: Analyzing Expression (h)

The expression is \(-abc\). From Step 3, \(bc < 0\). Now \(-abc\) involves multiplying this by \(-a\) where \(-a\) is negative (Step 1). A negative times a negative yields a positive. Therefore, the sign of \(-abc\) is positive.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (-) before the number. Negative numbers are used to signify a deficit, loss, or below-zero measurement. For example, if the temperature is -5°C, it means it is 5 degrees less than 0°C.

- **Negative of a Positive Number**: If you take a positive number and apply a negative sign, it becomes negative. For instance, if you take a positive number like 3 and apply a negative sign, you get \(-3\).
- **Multiplying Negatives**: When you multiply a negative number by another negative number, the result is a positive number. This rule is a foundation in multiplication of real numbers.

Understanding negative numbers helps in solving expressions where flipping signs play a crucial role. For example, in our problem, \(-c\) becomes positive because \(c\) itself is negative.

Exploring Positive Numbers

Positive numbers are all numbers greater than zero. They are straightforward in terms of arithmetic operations. Generally, when dealing with expressions involving positive numbers, the results stay intuitive until mixed with negative numbers.

- **Properties**: The sum of two positive numbers is always positive. Also, multiplying positive numbers yields a positive result.

In expressions where a clearly positive number like \(a\) in the exercise is used, the role of positive numbers becomes more crucial when they interact with negative numbers such as in subtraction or multiplication. This is evident in the expression \(a - bc\) where subtracting a negative (as seen from \(bc\)) effectively leads to addition.

Determining the Sign of Expressions

The sign of an expression refers to whether the expression results in a positive or negative number. Analyzing the sign of each term in an expression helps to predict the overall sign. For example:

- **Subtraction and Addition**: With terms like \(a - b\), when \(a > b\), the result is positive. In contrast, if a lesser number is subtracted from a greater number, like in \(c - a\), it results in a negative outcome.
- **Multiplication**: If the signs of the multiplied numbers are opposite, as in \(b\) and \(c\), the result is negative. However, if both numbers have the same sign, the result is positive.

Expressions containing mixed signs require careful consideration of each part. The use of parantheses can enhance clarity by grouping numbers and operations. This allows for an easier analysis of the expression's sign, especially in more complex calculations. Observing how operations alter the sign is integral in constructing and simplifying expressions.