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


In the process of converting a fraction to a decimal, all remainders must be less than the denominator of the fraction. Because the list of possible remainders is finite, and because each subsequent step is always the same (brings down a 0, etc.), the remainders must eventually repeat. The fraction 2/7 had six remainders (the maximum) and repeated in six digits. Other examples: 1/11 repeats in two digits, 1/13 repeats in six digits, and 1/17 repeats in 16 digits.

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.

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