Decimal place value

In 52.307 the 3 is worth three tenths, the 0 holds the hundredths place empty, and the 7 is seven thousandths.

Is 0.45 bigger than 0.5?

Which is bigger, 0.45 or 0.5? The quick answer is 0.45. Forty-five beats five, after all. That reasoning is fine for whole numbers and wrong here. Write both numbers with the same number of places, 0.45 and 0.50, and it falls apart: fifty hundredths is more than forty-five hundredths, so 0.5 is the larger number. The extra digit in 0.45 does not make it bigger. It only makes it finer. To see why, start with what the point actually does.

The point is a fence between whole units and pieces

A price tag that reads 3.75 dollars tells you about two kinds of amount at once. The 3 to the left of the point counts whole dollars. The 75 to the right counts parts of a dollar, here 75 hundredths of one. The decimal point is the fence between whole units and pieces of a unit, and every digit means something different depending on which side of the fence it sits on and how far from it.

Whole-number places grow by ten as you move left: ones, tens, hundreds. Decimal places shrink by ten as you move right: tenths, hundredths, thousandths, ten-thousandths. Each place is worth one tenth of the place on its left. OpenStax points out a small detail that does a lot of work: the "th" at the end of a name means a fraction. One thousand is bigger than one, but one thousandth is smaller than one.

Fifty-two point three zero seven, or fifty-two and three hundred seven thousandths

In English, OpenStax gives a four-step recipe. Name the whole-number part, say "and" for the point, name the digits after the point as if they were a whole number, then name the place of the last digit. So 52.307 is "fifty-two and three hundred seven thousandths", and 0.04 is "four hundredths". In everyday speech people often just read the digits: "fifty-two point three zero seven". Korean always uses the digit-by-digit style after the point.

The last digit decides the denominator

The last digit tells you the denominator. 0.7 ends in the tenths place, so it is 7/10. 0.07 ends in the hundredths place, so it is 7/100. 0.25 is 25/100, which simplifies to 1/4. A decimal with a whole part becomes a mixed number: 4.6 is 4 6/10, or 4 3/5 in simplest form. This is also why the fraction bar below shows a decimal: with 10 equal parts, each part is exactly 0.1.

Shade 3 of the 10 parts and the reading says 0.3 and 30%. Shade 7 and it says 0.7. Switch the bar to 4 parts and shade 1: the reading is 0.25, the same as 25 hundredths.

Why 0.50 equals 0.5 but 0.05 does not

Zeros at the end of a decimal change nothing: 0.5, 0.50 and 0.500 are the same number, five tenths, written with more places. Zeros between the point and a digit change everything: 0.05 is five hundredths, ten times smaller than 0.5. A zero in front of the point, as in 0.5 rather than .5, is not required, but it makes the point hard to miss, so this course always writes it.

Putting 0.3, 0.29, 0.305 and 0.31 in order

A longer decimal is not a bigger decimal. Which is larger, 0.6 or 0.58? Give them the same number of places by writing a zero at the end: 0.60 and 0.58. Now compare them as whole numbers of hundredths: 60 is more than 58, so 0.6 is larger. To order 0.3, 0.29, 0.305 and 0.31, write them all with three places: 0.300, 0.290, 0.305, 0.310. The order from smallest is 0.29, 0.3, 0.305, 0.31.

Decimals are everywhere that measurement is. A cent is one hundredth of a dollar, which is why prices stop at two places. NIST lists 1,000 millimetres in a metre, so 0.001 metres is one millimetre and a length such as 1.375 metres is 1 metre and 375 millimetres.