Determining the complete set of real square roots of 100 is a fundamental exercise that reveals the symmetry of the number line and the definition of principal roots. While the arithmetic result of 100 is 10, the question of square roots asks which numbers, when multiplied by themselves, yield this specific value. The real number system provides exactly two solutions, and understanding why requires a clear distinction between the inverse operation of squaring and the standard output of a calculator function.
The Definition of a Square Root
To find all real square roots of 100, one must first understand the formal definition of a square root. A square root of a number \( x \) is any value that, when multiplied by itself, equals \( x \). Therefore, we are looking for all real numbers \( r \) such that \( r \times r = 100 \). This equation can be rewritten as \( r^2 = 100 \). Solving for \( r \) involves isolating the variable, which leads to \( r = \pm \sqrt{100} \). The \( \pm \) symbol is critical here, as it indicates that both a positive and a negative solution exist.
The Positive and Negative Solutions
Breaking down the equation \( r^2 = 100 \), we can test the obvious candidates. Multiplying 10 by 10 results in 100, establishing 10 as a valid root. However, multiplication rules state that two negative numbers multiplied together yield a positive product. Consequently, multiplying -10 by -10 also results in positive 100. Thus, the two distinct real numbers that satisfy the condition are 10 and -10. There are no other integers, rational numbers, or irrational numbers that meet the criteria within the real number system.
Number | Calculation | Result
10 | 10 × 10 | 100
-10 | -10 × -10 | 100
The Principal Square Root vs. All Roots
A common point of confusion arises from the notation used in mathematics. The radical symbol \( \sqrt{} \) specifically denotes the principal square root, which is always the non-negative root. When a calculator evaluates \( \sqrt{100} \), it returns 10, not -10. This is a convention to ensure functions are well-defined and singular. However, the equation \( x^2 = 100 \) has two solutions. Therefore, the complete answer for "find all real square roots of 100" must include both the principal root (10) and its additive inverse (-10).
Graphical Interpretation
Visualizing the problem on a graph provides further clarity. The function \( y = x^2 \) is a parabola opening upwards with its vertex at the origin. Drawing a horizontal line at \( y = 100 \) intersects the parabola at exactly two points. These intersection points correspond to the x-values where the square of the input equals 100. The coordinates of these points are (10, 100) and (-10, 100), visually confirming that there are two real solutions, one positive and one negative, equidistant from zero.
