Comparing and Ordering Decimals to Thousandths

After the decimal point the columns are tenths, hundredths, thousandths, each one a tenth of the one before, exactly as the whole-number columns are. Four point zero seven two has four ones, no tenths, seven hundredths and two thousandths. Zeros on the right end rename without changing anything: three tenths is thirty hundredths is three hundred thousandths, so zero point three, zero point three zero and zero point three zero zero are all the same number. A zero in the middle is different — zero point zero three is three hundredths, ten times smaller than three tenths.

To compare decimals, pad them with zeros so they all have the same number of places, and then read them as whole numbers of the smallest unit. Zero point one becomes one thousand ten-thousandths; zero point zero eight becomes eight hundred; zero point zero zero eight nine is eighty-nine. Now the order is obvious, and only zero point one is greater than zero point zero eight. The shortest number is the largest. Or walk in from the left one column at a time — tenths first — and stop at the first column that differs.

Ordering a list is the same move on every number at once. Pad them all to the longest length, read them as whole numbers of that unit, sort, and then write the answer in the original forms. Zero point five, zero point four five and zero point four zero five become five hundred, four hundred and fifty and four hundred and five thousandths, so the order is zero point four zero five, zero point four five, zero point five. Numbers with wholes work the same: one point zero five, one point five, one point one five become one hundred and five, one hundred and fifty, one hundred and fifteen hundredths.

Two decimals that look next to each other always have room between them. Between zero point three and zero point three one: pad to thousandths, three hundred and three hundred and ten, and there are nine whole numbers between — zero point three zero one up to zero point three zero nine. Halfway is zero point three zero five, which is also what you get by adding the two and halving. There is no next decimal after zero point three, because you can always add another place and find ten more slots.

The claim format with decimals: pad every number once, write the thousandths next to each, and every statement becomes a comparison of whole numbers. Zero point one, zero point zero nine and zero point zero eight nine are one hundred, ninety and eighty-nine thousandths. So zero point one is the largest, true; zero point zero eight nine is less than zero point zero nine, true; zero point zero nine less than zero point zero eight nine, false. Statements one and two. Quiz and exercises now.