Problem 13 - Fundamentals of Mathematics

Question

Suppose \(a\) is a positive number. Is \(-a\) positive or negative?

Step-by-Step Solution

Answer

-a is negative.

Step 1: Understanding the Given Information

We are given that \(a\) is a positive number. That means \(a > 0\).

Step 2: Recognizing Opposite Signs

The expression \(-a\) indicates the opposite of \(a\). This involves changing the sign of \(a\).

Step 3: Determining the Sign of \(-a\)

Since \(a\) is positive, \(-a\) becomes its opposite, which will be negative because multiplying a positive number by \(-1\) results in a negative number.

Key Concepts

Positive Numbers

Positive numbers are numbers that are greater than zero. They are marked with the positive sign "+", but often the sign is left unwritten because positivity is assumed when no sign is present. These numbers appear to the right of zero on a number line.
Some examples of positive numbers include:

• Whole numbers like 1, 2, 3, etc.
• Fractions like 1/2, 3/4.
• Decimals like 0.5, 2.75.

Positive numbers represent quantities that can be counted or measured, such as money you earn, the distance you run, or the time you spend studying.

Negative Numbers

Negative numbers are numbers less than zero. They carry a "-" sign in front of them, indicating their position to the left of zero on the number line. Negative numbers represent a reverse concept to positive numbers, typically suggesting a shortage, debt, or a position below a certain point.
Examples of negative numbers include:

• -1, -2, -3, which are whole numbers.
• -1/2, -3/4, representing fractions below zero.
• -0.5, -2.75, which are negative decimals.

They are crucial in mathematics because they help explain scenarios like debts in finance, temperatures below freezing, and depths below sea level.

Sign Change

Sign change involves switching a number from positive to negative, or vice versa. This is accomplished by multiplying the number by -1. When you change the sign of a positive number, it becomes negative, and changing the sign of a negative number makes it positive. Multiplying helps to illustrate this: if you have a positive number like 5, multiplying it by -1 results in -5. Similarly, -(-5) equals 5, bringing back the number to its positive form. This concept of sign change is essential in determining operations with numbers, especially when dealing with equations and inequalities in algebra.

Opposite Numbers

Opposite numbers are numbers that have the same absolute value but different signs. They are equidistant from zero on a number line but in opposite directions.
For example:

• The opposite of 3 is -3, and vice versa.
• For a fraction, such as 1/4, its opposite would be -1/4.
• If we have -7.8, its opposite is 7.8.

Finding the opposite number is useful when solving problems that require balancing equations or understanding balances in math. The idea shows the symmetry of numbers around zero and how any number plus its opposite equals zero.