A square root is a fundamental concept in mathematics. It is a value that, when multiplied by itself, gives us the original number. For example, 3 times 3 equals 9, so 3 is a square root of 9. We use the radical symbol, square root sign, to represent the principal or positive square root.
Perfect squares are numbers that result from multiplying an integer by itself. For example, 1 squared equals 1, 2 squared equals 4, 3 squared equals 9, and so on. The square roots of these perfect squares are the original integers we started with. So the square root of 1 is 1, the square root of 4 is 2, the square root of 9 is 3, and this pattern continues.
An important concept to understand is that every positive number actually has two square roots: one positive and one negative. For example, both 3 and negative 3 are square roots of 9, because 3 times 3 equals 9, and negative 3 times negative 3 also equals 9. However, when we use the radical symbol, it always represents the principal or positive square root only.
The square root function, written as f of x equals square root of x, is a fundamental mathematical function. Its domain includes only non-negative numbers, since we cannot take the square root of negative numbers in the real number system. The range is also non-negative, meaning all outputs are zero or positive. The graph shows a curved line starting at the origin and increasing gradually as x increases.
To summarize what we have learned about square roots: A square root finds the value that multiplies by itself to give the original number. Perfect squares have exact integer square roots. Every positive number has both positive and negative square roots, but the radical symbol represents only the positive one. The square root function works only with non-negative numbers.