A Peek Beyond the Point

Suppose a screw measures a little more than 2 cm but less than 3 cm on a ruler — how do you record its length exactly? A whole number like 2 cm or 3 cm isn't precise enough. The answer is to split the space between 2 and 3 into 10 equal parts, called tenths. If the screw's tip lands on the 7th of these 10 small marks past the 2 cm mark, its length is 2 units and 7 tenths, written as the mixed fraction 2 7/10 cm, and read as "two and seven-tenths centimetres."

This idea connects directly to something you already know: 10 ones make 1 ten, 10 tens make 1 hundred, and so on going up the place value chart. Splitting a unit into 10 tenths is simply the same pattern extended downward, below the ones place. One-tenth is the amount you get when a single unit is divided into 10 equal pieces — so 10 of these pieces put back together make exactly 1 whole unit again.

This isn't limited to length. Sonu measuring his lower arm as 2 7/10 units and his upper arm as 3 6/10 units can add the two lengths by adding units and tenths separately: (2 + 3) units and (7 + 6) tenths = 5 units and 13 tenths. But 13 tenths is more than a full unit — 10 of those tenths regroup into 1 more unit, leaving 3 tenths over. So the total length is 6 units and 3 tenths, i.e., 6 3/10 units.

Sometimes even tenths aren't fine enough. Splitting each one-tenth into 10 smaller parts gives hundredths — so there are 100 hundredths in a full unit. A length like 4 units, 4 tenths and 5 hundredths can be written as the mixed fraction 4 45/100 (since 4 tenths and 5 hundredths together make 45 hundredths), or using place value as 4 4/10 5/100.

Writing fractional place values with separate fraction symbols every time is clumsy, so we use a decimal point ('.') to mark where the whole-number part ends and the tenths-hundredths part begins. The quantity 4 units, 4 tenths and 5 hundredths is written compactly as 4.05 — but that would actually be a mistake. 4.05 means 4 units, 0 tenths, and 5 hundredths (reading digit-by-digit after the point: tenths place, then hundredths place). To write "4 units and 45 hundredths" (which is the same amount as 4 units, 4 tenths, 5 hundredths) in decimal form, you'd write 4.45, not 4.05 — the position of each digit after the point matters enormously.

So 70.5 means 7 tens, 0 units, 5 tenths (read as "seventy point five"), while 7.05 means 7 units, 0 tenths, 5 hundredths (read as "seven point zero five") — moving the decimal point changes which place value each digit belongs to, even though both numbers use the same digits 7, 0 and 5.

The decimal place value chart extends the whole-number chart to the right of the ones place: Hundreds, Tens, Ones . Tenths, Hundredths, Thousandths — and just like with whole numbers, each place is exactly 10 times smaller than the place to its left. This consistent '10 times' pattern is exactly why the system is called the decimal system (from the Latin decem, meaning ten).

A natural question: does writing extra zeroes after a decimal change its value? Comparing 0.2, 0.20 and 0.200 using place value — all three represent exactly 2 tenths, just written with extra trailing zeroes in the hundredths and thousandths places, which contribute nothing. So 0.2 = 0.20 = 0.200. But 0.02 is a completely different quantity — it means 2 hundredths, not 2 tenths, so 0.2 is not equal to 0.02. Trailing zeroes placed after every meaningful digit are harmless; zeroes inserted in between digits change the value entirely.

This same care is needed when comparing two decimals of different lengths, like 11.01 and 11.10. Compare digit-by-digit from the most significant place: both have 1 ten and 1 one, so move to the tenths place — 11.10 has 1 tenth while 11.01 has 0 tenths. Since 1 is greater than 0 at this shared place, 11.10 is the larger number, even though 11.01 has a non-zero digit further to the right (in the hundredths place) that 11.10 doesn't.

Length, weight and money in India are all built on the same base-10 pattern as decimals, which makes conversion between units quick. Since 1 cm = 10 mm, 1 mm is one-tenth of a centimetre: 1 mm = 0.1 cm. So 12 mm, being 1 cm and 2 mm, converts to 1.2 cm. Going the other way, 5.6 cm (5 cm and 0.6 cm) converts back to 56 mm, since each 0.1 cm is worth 1 mm.

The same logic applies between centimetres and metres, since 1 m = 100 cm, making 1 cm one-hundredth of a metre: 1 cm = 0.01 m. So 15 cm converts to 0.15 m, and more generally, any length in cm can be converted to metres just by shifting the decimal point two places to the left.

Weight works the same way: since 1 kg = 1000 g, each gram is one-thousandth of a kilogram, so 254 g = 0.254 kg. And money follows the identical pattern: since 1 rupee = 100 paise, each paisa is one-hundredth of a rupee, so 75 paise = 0.75 rupees. Whenever a 'smaller' unit fits a fixed number of times into a 'bigger' one — 10, 100, or 1000 times — decimal conversion is simply a matter of moving the decimal point by the matching number of places.

Adding or subtracting decimals uses the same standard procedure as whole numbers, as long as the decimal points are aligned so that tenths add to tenths, hundredths add to hundredths, and so on. For example, adding 75.345 and 86.691 place by place from the right (thousandths, hundredths, tenths, ones, tens) and carrying wherever a column totals 10 or more gives 162.036. Subtraction works the same way, borrowing across the decimal point exactly as you'd borrow across the ones and tens columns in a whole number.

Comparing decimals that are close together needs the same digit-by-digit approach as before. Among 0.9, 1.1, 1.01 and 1.11, which is closest to 1? Checking the distance of each from 1: 0.9 is 0.1 away, 1.1 is 0.1 away, 1.01 is only 0.01 away, and 1.11 is 0.11 away. Since 0.01 is the smallest distance, 1.01 is the closest to 1 among these four numbers — even though 1.1 looks closer at first glance because it has fewer digits.

One more comparison worth practising: is 1.009 or 1.090 greater? Both numbers have 1 one and 0 tenths, so the tie is broken at the hundredths place: 1.009 has 0 hundredths, while 1.090 has 9 hundredths. Since 9 is greater than 0 at this first differing place, 1.090 is greater than 1.009, even though 1.009 has a non-zero digit in the thousandths place that 1.090 doesn't.