You are correct. An expanded number is a number written out in full. The powers of ten, represented by places to the right or left of the decimal point, are the same for every number written in the decimal system. This is why numbers can be written in the normal, or unexpanded, form, with only coefficients showing.
In the decimal number system, each “place” to the right or left of the decimal point represents some power of ten. Each place is occupied by a coefficient which tells how many times that particular power of 10 is to be added into the number.
The order in which the powers of ten are represented is fixed. The first place to the left of the decimal point is occupied by the coefficient of 100. Since 100, or “ten to the power zero” equals one, the coefficient in the first place to the left of the decimal point tells how many “ones” are to be added into the number.
Similarly, the second place to the left of the decimal point is occupied by the coefficient of 101. This coefficient tells how many tens are in the number. The number in the third place to the left of the decimal point is the coefficient of 102 and tells how many hundreds there are in the number.
Thus the number 153 can be written in expanded form in the following way:
1(102) + 5(101) + 3(100).
The numbers 1, 5, and 3 are the multipliers or coefficients of 10 to the 2nd power, 10 to the 1st power, and 10 to the zeroeth power, respectively.
In decimal fractions, each place to the right of the decimal point also represents a power of ten - in this case, a negative power of ten. The first place to the right of the decimal point is occupied by the coefficient of 10 − 1, or, as it may be written, (1)/(101). (1)/(101), of course, equals (1)/(10). So the coefficient of 10 − 1 tells how many tenths there are in the decimal fraction.
Similarly, the number which occupies the second place to the right of the decimal point is the coefficient of 10 − 2. 10 − 2, as we have shown, is the same as (1)/(102), or (1)/(100). So this coefficient tells how many hundredths are in the decimal fraction.
The expanded form of the decimal fraction .594, then, might be written as follows:
5(10 − 1) + 9(10 − 2) + 4(10 − 3)
or,
(5)/(10) + (9)/(100) + (4)/(1000).
Thus the coefficients of these negative powers of ten are the digits of the decimal fraction.
Because coefficients tell us which power of ten they multiply by their position on the left or right side of the decimal point, they must always be written in the proper order. If someone decided to change the order of the coefficients, they would mean something entirely different than they do: 325 is not 532!
We are so accustomed to using coefficients in our decimal system that we don’t stop to think about it. We see the number “532” as a unit made up of three digits. However, we know that this means five hundred(s) thirty (3 tens) and two (2 ones).
The reason for explaining the expanded form of a decimal number at this time will become apparent in the next lesson. The key to the arithmetic of computers is in understanding that the “numbers” we toss around so casually are in fact made up of coefficients of particular powers of the base of one number system.
Now please turn to the Self-Test Questions on Lesson 5, in the Appendix.
