MTH 231 Section 3.5 Nondecimal Positional Systems

Overview As previously discussed, our numeration system is a decimal (or base-ten) system, likely because we have ten fingers. It is, however, entirely possible to use the principles of our decimal system and expanded notation to explore systems with bases other than ten.

Important! The base of a numeration system tells you, among other things, how many digits there are in that system: Base-ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 So, a base-six system will have six digits (0, 1, 2, 3, 4, 5) and a base-five system will have five digits (0, 1, 2, 3, 4).

Equally Important! Expanded notation of our base-ten system can be generalized to other bases: We say that the 7 is in the hundreds place, the 9 is in the tens place, and the 2 is in the ones place.

One More Important!! Words such as hundred, thousand, million, and so on, can not be used in other base systems. They are strictly reserved for use in a base-ten system.

Quinary (Base-five) The places are based on powers of five rather than ten:

An Example Not “three thousand one hundred four” but rather “three one zero four, base five”

Senary (Heximal or Base-six) The places are based on powers of six rather than ten:

Converting From Base-ten 1.List all of the places, starting with the ones place (all base systems will have a ones place) and going up to the largest place less than the decimal number. 2.Divide the decimal number by that largest place. Write the quotient in that place. 3.Now, divide the remainder by the next place, unless the remainder is less than the next place (in which case you write 0 in that place and move on). 4.Repeat the process until every place, all the way down to the ones place, has a digit.

Examples Convert each of the following to base-five and base-six: Your birth year

Converting To Base-ten Use the idea of expanded notation to give each digit a value derived from its place:

Examples Convert each of the following to base-ten six five

Addition and Subtraction Work pretty much the same way in base-five and base-six as they do in base-ten (exchanges, borrowing, carrying) Remember: you cannot use digits that do not exist in the system!

Try These: 314 five five 314 five – 231 five 431 six six 431 six – 213 six