The long division of 2÷7 is more complicated. The remainder at the seventh step is 2, which is where the first step began. Because there will always be another 0 to “bring down” in the next place, the sequence of remainders (2, 6, 4, 5, 1, 3) will repeat, as will the digits 285714 in the quotient. Thus, a repeating decimal, where the horizontal bar is used to indicate which digits repeat.
The process of using long division to obtain the decimal representation of a fraction will always be like one of the above cases: Either the process will stop or it will cycle through some sequence of remainders. So the decimal representation of a rational number must be either a repeating or a terminating decimal. Thus a nonrepeating decimal cannot be a rational number and there are many such numbers, such as p and
Understanding a mathematical idea thoroughly requires that several possible representations be available to allow a choice of those most useful for solving a particular problem. And if children are to be able to use a multiplicity of representations, it is important that they be able to translate among them, such as between fractional and decimal notations or between symbolic representations and the number line or pictorial representations.
Addition is an idea—an abstraction from combining collections of objects or from joining lengths. Carrying out the addition of two numbers requires a strategy that will lead to the result. For single-digit numbers it is reasonable to use or imagine blocks or cookies, but for multidigit numbers you need something more efficient. You need algorithms.