The square root is an important mathematical concept that is usually encountered once the basic rules of arithmetic have been mastered.
To appreciate this mathematical concept let’s first, let’s ask a question. Is it possible to multiply two equal numbers together to produce a third? This is a no brainer. we all know how to do this using the elementary rules of multiplication.
Now, lets look at the opposite question. Can we find two equal numbers that multiply together to produce a specified number? This is a much harder problem which leads to the idea of the square root.
Mathematicians define a square root a number which when multiplied by itself makes the specified number. We can write this in a shorthand using the symbol √. To show how this works let’s write √ 4. We know that 2 times 2 equals 4 so √ 4 = 2 . (As an aside the √.symbol was introduced by Rudolf in 1525. It is probably derives from a badly written r which was used to denote a radical or square root in the Middle Ages).
The square root often comes up in practical calculations and in engineering formula. A typical problem might involve finding the length of walls required to enclose a certain floor space. For example, if we wished to enclose a floor space of 9 square metres we would need to build walls √.9 or 3 metres long. The square root also crops up in the formula to calculate the length third side of a right angle triangle when the length of the other two sides are known using Pythagoras’ theorem.
Square roots were widely studied to the ancient Greeks. The Greeks realised that every number has two square roots. Both have the same value but one is positive and the other is negative. This arises because a negative number multiplied by a negative number is a positive number. Much later in the time
One of the strange things about square roots is that it is impossible to express most of them exactly as fractions or decimals. Numbers that can not be expressed as a fraction are called irrational numbers. When the Greeks discovered that square roots could be irrational it was disconcerting. Until that time the Greeks believed that all numbers were rational.
For a long time the question of where a negative number has a square root was left unresolved. In 1637 Rene Descartes solved this problem by introducing the notation i to represent the imaginary number √ (- 1). His notation allowed a negative number to be factorised into two parts from which the square root could be derived. For example Descarters would write √ (- 9) as√(-1).√9 or 3i. These numbers sometimes come up in wave theory and in electrical engineering.
The square root is a very interesting mathematical concept which has led to several very important discoveries such as irrational and imaginary numbers. It is one of the first areas of advanced mathematics that is encountered in the classroom.
