Comparing decimals on the number line

Decimals on a number line

The decimal 1.4 is 1 unit and 4 tenths, so it lies between 1 and 2. To mark it, split the gap from 1 to 2 into 10 equal parts and count 4 parts from 1. The marks are 1.1, 1.2 and so on up to 1.9, and the tenth mark is 2.

Every small gap can be split again. Between 1 and 1.1 there are 9 marks in hundredths: 1.01, 1.02 and so on up to 1.09. On a line from 5 to 5.4 drawn in hundredths, a point one step before 5.1 is 5.09, a point three steps after 5.1 is 5.13, and a point one step after 5.3 is 5.31.

Zooming in

To find a number with three decimal places, such as 4.185, zoom in one place at a time. On a line from 0 to 10, 4.185 lies between 4 and 5. Magnify that unit and split it into tenths: the number lies between 4.1 and 4.2. Magnify that tenth and split it into hundredths: it lies between 4.18 and 4.19. One more split into thousandths, and 4.185 is the middle mark. The same steps locate 9.876 (between 9 and 10, then 9.8 and 9.9, then 9.87 and 9.88) or 0.407.

Not every number line is split into tenths of a unit. If the gap from 5 to 10, which is 5 units, is split into 10 equal parts, then each part is half a unit. The marks are 5.5, 6, 6.5 and so on, so the 3rd mark after 5 is 6.5 and the 5th is 7.5. Always work out the size of one division before reading a mark.

The zero dilemma

Sonu says 0.2 can be written as 0.20 or 0.200. Zara worries that zeros on the right might change the value. A place value table settles it: 0.2, 0.20 and 0.200 all have 2 in the tenths place and nothing else, so they are equal. But 0.02 has its 2 in the hundredths place and 0.002 in the thousandths place, so they are smaller: 0.2 is the largest and 0.002 the smallest.

So in the list 4.5, 4.05, 0.405, 4.050, 4.50, 4.005 and 04.50, the equal ones are 4.5 = 4.50 = 04.50 and 4.05 = 4.050, while 0.405 and 4.005 stand alone.

Comparing digit by digit

Which is larger, 6.456 or 6.465? Both have 6 units. Both have 4 tenths. In the hundredths, the first has 5 and the second 6, so 6.465 is larger, and we can stop. The digits that follow can never change this: whatever comes after the hundredths is worth less than one hundredth in all, so it cannot make up a gap of a full hundredth. This is the same rule used for whole numbers: compare the most significant digits first.

Line up the points before comparing. 13.800 is greater than 3.81, because it has a tens digit and 3.81 has none. 1.090 is greater than 1.009, because they first differ in the hundredths, where 9 beats 0. A number with more digits is not always larger: 2.05 < 2.5.

Which decimal is closest?

Which of 0.9, 1.1, 1.01 and 1.11 is closest to 1? In order they are 0.9 < 1 < 1.01 < 1.1 < 1.11. The nearest neighbours of 1 are 0.9, which is 1/10 away, and 1.01, which is only 1/100 away, so 1.01 is the closest. In the same list, 1.1 is closest to 1.09, being only 0.01 away from it.

Why it matters

Prices, race times and measurements are compared as decimals every day. What each place means is explained in tenths and hundredths, and how to work out sums and differences in adding and subtracting decimals. Answers to every question, including the digit puzzles, are in the NCERT solutions for Chapter 3; see all of the Chapter 3 notes.