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Integers (Latin, ‘unbroken’) are the first step onwards from the natural numbers. They arose much later, not really being accepted by mathematicians until after the Middle Ages, and they extend the natural numbers by the concept of owing. In the natural numbers, many equations of the form a + b = c can be solved. For example, the solution of a + 2 = 4 is a = 2. However, there is no solution of the equation a + 4 = 2; a ends up ‘owing’ 2. In the integers, the solution of such equations always exists; in this case, it is written -2.
If the natural numbers are viewed as an infinite line 0 1 2 3 and so on, the integers are an infinite line extended in both directions rather than only one (the natural numbers have a beginning, 0, while the integers do not), so that we have … -3 -2 -1 0 1 2 3 …, the ‘…’ representing indefinite continuation of the sequence in both directions.
In the integers, then, there is a solution to every equation involving addition of integers. However, when multiplication is brought in, there are once again equations which do not have solutions. For example, x × 2 = 4 has a solution, but x × 4 = 2 does not. The integers were therefore not powerful enough, and so the rational numbers were constructed. SMcL |
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