Problem 13 - Elementary Algebra

Question

Suppose that \(a\) is a positive number. What type of number is \(-a\) ?

Step-by-Step Solution

Answer

Answer: If \(a\) is a positive number, then \(-a\) is a negative number.

Step 1: Definition of a positive number

A positive number is any real number greater than zero. In this exercise, we are given that \(a\) is a positive number, so we know that \(a > 0\).

Step 2: Negating a positive number

When we negate a positive number, we obtain its opposite, which is a negative number. This means that if \(a\) is positive, \(-a\) is negative.

Step 3: Identifying the type of \(-a\)

Since \(a\) is a positive number and we have negated it, \(-a\) is a negative number. That's the answer to the exercise. \(-a\) is a negative number.

Key Concepts

Positive Numbers

Positive numbers are found everywhere in mathematics, representing quantities that are greater than zero. These numbers play a fundamental role in various applications, from counting objects to expressing measurements such as distance, weight, and temperature in a positive context.

When you see a number without a sign, it is usually understood to be positive. For example, the number 3 is a positive number, as is 100. These numbers fall to the right of zero on the number line, a visual representation of the set of all real numbers.

• A positive number plus another positive number will always result in a positive sum.
• Multiplying two positive numbers yields a positive product.
• When a positive number is divided by another positive number, the quotient is positive.

Understanding positive numbers helps lay the foundation for all operations involving real numbers.

Real Numbers

The set of real numbers includes all the numbers that can be found on the number line. This range includes not just the positive and negative integers, but also fractions, decimals, and irrational numbers like \( \pi \) and \( \sqrt{2} \).

• Positive numbers and zero are part of the non-negative real numbers.
• Negative numbers, those to the left of zero on the number line, are also real numbers.

The concept of real numbers is essential as it allows for the measurement and representation of continuous quantities. Moreover, when you perform operations (like addition or multiplication) with two real numbers, the result is also a real number, maintaining the 'closure' property of the set.

Negating a Positive Number

Negating a positive number means taking its additive inverse, which results in a negative number. This operation essentially 'flips' the number's position on the number line about zero.

If you take a positive number, say +5, and negate it, you will get -5. In mathematical terms, if \( a > 0 \), then \( -a < 0 \).

This process is crucial in solving equations, especially when isolating variables. When negating a positive number, remember that you are finding the number that, when added to the original, results in zero (the very definition of additive inverses).

• Negating a positive number twice will take you back to the original positive number, indicating that negation is a reversible operation.
• The absolute value of a number remains the same whether the number is positive or negated, emphasizing that negation affects direction but not magnitude.