5.5 Decimal Structure and Representation

Decimal Structure and Representation covers the base-ten positional system used to write real numbers with a decimal point, describing the specific place values assigned to each digit's position, the whole and fractional parts a decimal splits into, the conventions for reading and writing decimals, the handling of trailing zeros and signs, decimal placement and comparison on the number line, the distinction between terminating and repeating decimals, and the practice of rounding a decimal to a limited number of digits.

Decimal Place-Value Positions

Decimal place-value positions assign each digit in a decimal a specific value based on its distance from the decimal point, with positions to the left representing ones, tens, hundreds, and so on, and positions to the right representing tenths, hundredths, thousandths, and so on, each position worth ten times more or ten times less than its neighbor.

Whole and Fractional Decimal Parts

The whole part of a decimal is the portion of digits to the left of the decimal point, representing a value of one or greater, and the fractional part is the portion of digits to the right of the decimal point, representing a value less than one. In 3.14, the whole part is 3 and the fractional part is 0.14.

Decimal Reading and Writing

Decimal reading and writing follows the convention of stating the whole part first, followed by "and" or "point," followed by the fractional digits, often named according to the place value of the final digit, such as reading 3.14 as "three point one four" or "three and fourteen hundredths."

Equivalent Trailing Zeros

Equivalent trailing zeros are zeros appended to the end of a decimal's fractional part that do not change its value, such as 0.5, 0.50, and 0.500 all representing the identical value, since each additional trailing zero simply adds a place value of zero contribution.

Positive and Negative Decimals

Positive and negative decimals extend the sign conventions of whole numbers to decimal values, with a positive decimal lying to the right of the origin and a negative decimal, marked with a leading minus sign, lying to the left of the origin on the number line.

Decimal Number-Line Placement

Decimal number-line placement locates a decimal value at its position between two consecutive marks on the number line, determined by subdividing the interval between those marks according to the decimal's fractional digits, with finer subdivisions needed for decimals with more fractional digits.

Decimal Comparison

Decimal comparison determines which of two decimals is greater by comparing their whole parts first, and if those are equal, comparing corresponding fractional digits from left to right until a difference is found, following the same left-to-right numerical order principle established for the real numbers generally.

Terminating Decimal Structure

A terminating decimal has a fractional part consisting of a finite number of digits, ending after some fixed position with no further nonzero digits following, such as 0.75.

Repeating Decimal Structure

A repeating decimal has a fractional part in which one or more digits repeat infinitely in a fixed pattern, such as 0.333..., often marked with a bar over the repeating digits to indicate the pattern continues without end.

Repeating Block Identification

Repeating block identification determines the exact sequence of digits that repeats within a repeating decimal, such as identifying the repeating block "18" in the decimal expansion of 2/11, which equals 0.181818...

Exact and Approximate Decimal Values

Exact and approximate decimal values distinguishes a decimal representation that captures a number's value completely, such as a terminating decimal or a fully marked repeating decimal, from a decimal representation that has been cut off or rounded and therefore only estimates the true value.

Decimal Rounding

Decimal rounding replaces a decimal with a shorter approximation by examining the digit immediately following the desired rounding position and adjusting the final retained digit up by one if that following digit is 5 or greater, or leaving it unchanged if that following digit is less than 5. Rounding 3.14159 to two decimal places produces 3.14, since the digit following the hundredths position is 1, which is less than 5.

Together, these structural concepts describe how decimals are built from positional place values, split into whole and fractional parts, written and compared according to consistent conventions, and classified as terminating or repeating, with rounding providing the standard technique for approximating a decimal value to a manageable number of digits when its exact form is unnecessary or impractical.