Problem 43 - Algebra 1: Concepts and Skills

Question

Determine whether the number is a perfect square. $$ -5 $$

Step-by-Step Solution

Answer

No, -5 is not a perfect square.

Step 1: Understand what a perfect square is

A perfect square is a number that can be expressed as the product of an integer by itself.

Step 2: Observe the given number

The provided number is -5.

Step 3: Determine if the number can be a perfect square

Neither positive nor negative numbers when squared can produce a negative result. Thus, a negative number, like -5 in this case, cannot be a perfect square.

Key Concepts

Integer

An integer is a whole number that can be either positive, negative, or zero. This means integers include numbers like -3, 0, and 7. Integers do not have any decimal or fractional parts— they are complete numbers.

An important property of integers is that when you multiply any integer by itself, the result is also an integer. For example, multiplying 4 by 4 gives 16, which is still an integer.

• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Non-negative integers: 0 plus all positive integers

Understanding integers is crucial because they help form the basis of understanding other math concepts like perfect squares.

Negative Number

Negative numbers are numbers less than zero. They are represented with a minus sign (-), like -1, -5, or -10. Negative numbers are generally used to represent loss, debt, or any condition below a defined zero point.

It's important to know how negative numbers behave when operations are applied:

• Adding a negative is equivalent to subtracting.
• Multiplying two negative numbers results in a positive number.
• Multiplying a positive number by a negative number results in a negative number.

When concerning perfect squares, negative numbers cannot be squared to yield a negative result. For example, squaring -3 gives 9, not -9, affirming that no real number squared can ever yield a negative number like -5.

Square Number

A square number, or a perfect square, results when an integer is multiplied by itself. This type of number is called a "square" because it can be represented as a square shape with integer sides in geometry.

Here are some examples to make this clearer:

• (2 imes 2) = 4, which is a perfect square.
• (-3 imes -3) = 9, which is also a perfect square.
• 0 is a special case where (0 imes 0) = 0, also a perfect square.

Understanding square numbers is particularly useful in geometry, algebra, and various mathematical applications. In this context, it is essential to remember that square numbers are always non-negative. Therefore, negative values like -5 cannot be perfect squares because you cannot multiply two identical integers to achieve a negative number.