Problem 15 - Elementary Algebra

Question

Suppose we do not know the sign of the number \(m\). Can we say that \(-m\) is positive, negative, or that we do notknow ?

Step-by-Step Solution

Answer

Answer: Without knowing the sign of m, we cannot determine the sign of -m. It could be positive, negative, or zero, depending on the value of m.

Step 1: Consider the multiplication by -1 rule

Multiplication of any number by -1 will change its sign. So, if the number is positive, it becomes negative, and if the number is negative, it becomes positive.

Step 2: Case 1: m is positive

If m is a positive number, then -m will be negative. For example, if m = 3, then -m = -3.

Step 3: Case 2: m is negative

If m is a negative number, then -m will be positive. For example, if m = -3, then -m = 3.

Step 4: Case 3: m is zero

If m is zero, then -m will also be zero, which is neither positive nor negative.

Step 5: Conclusion

Since we don't know the sign of m, we cannot determine the sign of -m without more information. It could be positive, negative, or zero, depending on the value of m.

Key Concepts

Sign of a Number

In mathematics, the sign of a number determines whether it is positive or negative. Knowing a number's sign helps us classify it within the number line.
Positive numbers lie to the right of zero and are greater than zero. Negative numbers lie to the left of zero and are less than zero. Zero itself is a unique number which is neither positive nor negative.

• Example: 5 is a positive number, while -5 is a negative number.
• The sign tells us about the orientation on the number line: Positive (more than zero) or Negative (less than zero).

Multiplication by -1

An interesting property of multiplication in algebra is how the sign of a number changes when it is multiplied by -1. When any number is multiplied by -1, its sign reverses. So, a positive number becomes negative, and a negative number becomes positive.
For example, multiplying 7 (a positive number) by -1 results in -7.
On the other hand, multiplying -9 (a negative number) by -1 results in 9.

• Positive number × (-1) = Negative number
• Negative number × (-1) = Positive number

This property is quite essential in solving algebraic equations and inequalities.

Positive and Negative Numbers

Positive and negative numbers are foundational concepts in algebra. They help us express a wide range of real-world quantities, like temperature, altitude, and financial data. Positive numbers are greater than zero and represent values like profit and increase. Negative numbers are less than zero and represent values like debt and decrease.
Visualizing these numbers on the number line can help:

• Numbers to the right of zero are positive.
• Numbers to the left of zero are negative.

Understanding these differences is crucial for operations such as addition, subtraction, and multiplication of numbers.

Zero as Neither Positive nor Negative

Zero is a unique number in math. It acts as a divider on the number line between positive and negative numbers, but it is neither positive nor negative itself. When we multiply zero by any number, the result is always zero, maintaining its neutrality.
This makes it a bit of a special case in algebraic calculations. For instance, when a variable is zero, its product with -1 results in zero: multiplying 0 by any number or sign won't change it from zero.

• Zero acts as the neutral point between positives and negatives.
• Zero multiplied by any number remains zero.

This neutrality must be remembered when considering operations that involve zero.