Ganita Prakash I · Chapter 3 — A Peek Beyond the Point

Imagine you are fixing a toy with a screw. You pick a screw from the box and it does not fit. Another screw that looks almost the same fits perfectly. When you put both screws against a ruler, one ends a little after the 2 cm mark and the other a little after the 3 cm mark. Saying ‘between 2 cm and 3 cm’ is not good enough. We need a smaller unit.

Splitting a unit into ten parts. On a centimetre ruler, the gap between two numbers is cut into 10 equal small parts. Each small part is one-tenth of a centimetre, written 1/10 cm. Ten of these tenths make one whole centimetre: 1/10 + 1/10 + … (ten times) = 10/10 = 1.

Reading a length. If a screw ends at the 7th small mark after 2, its length is 2 cm and 7 tenths of a centimetre. We write it as a mixed number, 2 7/10 cm, and say ‘two and seven-tenths centimetres’. This means 2 + 7 × 1/10.

Two ways to say the same length. Since 10 tenths make 1 unit, 2 units are 20 tenths. So 2 7/10 is also 27 tenths, written 27/10. Both describe the same point on the ruler.

Comparing lengths in tenths. To put lengths in order, first look at the whole units, then at the tenths. For example 9/10 is less than 1, so it is smaller than 1 7/10. And 130/10 is the same as 13, so it is a little less than 13 1/10. Turning everything into tenths is a safe way to compare: 1 7/10 = 17 tenths and 6 7/10 = 67 tenths, so 1 7/10 is smaller.

Where we use tenths. A tailor measuring cloth, a doctor reading a thermometer and an athlete timing a race all need parts of a unit. Whenever whole units are too rough, we split each unit into 10 smaller equal parts and count those parts.

A girl measures her arm in two parts. Her lower arm is 2 7/10 units and her upper arm is 3 6/10 units. How long is the whole arm? We need to add numbers that have tenths.

Method 1: add units and tenths separately. Units: 2 + 3 = 5. Tenths: 7 + 6 = 13 tenths. But 13 tenths is 10 tenths and 3 tenths, which is 1 unit and 3 tenths. So the total is 5 + 1 + 3/10 = 6 3/10 units. This is just like carrying in ordinary addition: when the tenths reach 10, we exchange them for one unit.

Method 2: change both to tenths. 2 7/10 = 27 tenths and 3 6/10 = 36 tenths. 27 + 36 = 63 tenths = 63/10. Now 63 tenths is 60 tenths and 3 tenths, which is 6 units and 3 tenths: 6 3/10. Both methods agree.

Subtraction. A hand is 12 4/10 units long and the palm is 6 7/10 units. How long is the longest finger? We find 12 4/10 − 6 7/10. We cannot take 7 tenths from 4 tenths. So we break one unit of 12 into 10 tenths: 12 4/10 becomes 11 units and 14 tenths. Now 14 − 7 = 7 tenths and 11 − 6 = 5 units. The finger is 5 7/10 units.

Check by tenths. 12 4/10 = 124 tenths and 6 7/10 = 67 tenths. 124 − 67 = 57 tenths = 5 7/10. The same answer.

Patterns with tenths. Look at the list 4, 4 3/10, 4 6/10, … Each term is 3 tenths more than the one before, so the next terms are 4 9/10, 5 2/10, 5 5/10 and 5 8/10. Notice how 4 9/10 + 3/10 = 4 12/10 = 5 2/10: the tenths passed 10 and became a unit. In a decreasing list such as 11 5/10, 10 4/10, 9 3/10, … each step takes away 1 1/10.

Tips.

• Start with the tenths, just as you start with ones in ordinary addition.
• If the tenths add up to 10 or more, exchange 10 tenths for 1 unit.
• If you cannot subtract the tenths, break one unit into 10 tenths.
• Check with the other method when you have time.

A strip of paper is 8 9/10 units long. Fold it in half. Its length is now between 4 4/10 and 4 5/10, but the tenth marks cannot tell us exactly where. We need a smaller unit again.

Splitting a tenth. Cut each tenth into 10 equal parts. Since a unit has 10 tenths and each tenth now has 10 small parts, a unit has 10 × 10 = 100 of them. Each one is a hundredth, 1/100 of a unit.

The folded paper. Half of 8 9/10 is half of 89 tenths, which is 890 hundredths ÷ 2 = 445 hundredths. That is 4 units, 4 tenths and 5 hundredths, written 4 4/10 5/100, or 4 45/100.

Three ways to write one length. Because 1 tenth = 10 hundredths:

Comparing. Compare the units first, then tenths, then hundredths. Which is longest: 3/10, 3/100 or 33/100? Change them all to hundredths: 30/100, 3/100 and 33/100. So 33/100 is longest and 3/100 is shortest.

Adding with hundredths. Find 15 3/10 4/100 + 2 6/10 8/100. Units: 15 + 2 = 17. Tenths: 3 + 6 = 9. Hundredths: 4 + 8 = 12 = 1 tenth and 2 hundredths. Now tenths: 9 + 1 = 10 = 1 unit. Total: 18 units, 0 tenths, 2 hundredths = 18 2/100. You can check by hundredths: 1534 + 268 = 1802 hundredths = 18 2/100.

Subtracting with hundredths. Find 25 9/10 − 6 4/10 7/100. The first number has no hundredths, so break 1 tenth into 10 hundredths: 25 8/10 10/100. Now hundredths 10 − 7 = 3, tenths 8 − 4 = 4, units 25 − 6 = 19. Answer: 19 4/10 3/100, that is 19 43/100.

Same as whole-number sums. In 483 + 268, ones are exchanged for tens and tens for hundreds. Here hundredths are exchanged for tenths and tenths for units. The rule ‘ten of one place make one of the next place’ is the same.

In the number 281, the 2 means 2 hundreds, the 8 means 8 tens and the 1 means 1 one. Each place is 10 times the place on its right, and 1/10 of the place on its left. Can this pattern go further to the right of the ones place?

Extending the places. Divide 1 by 10 and we get one-tenth. Divide again and we get one-hundredth, then one-thousandth, and so on. So the places now run: … thousands, hundreds, tens, ones, tenths, hundredths, thousandths …, and every step to the right divides by 10. This is the decimal system, named after the Latin word for ten (it is related to the Sanskrit word for ten, daśa).

The need for a point. If we wrote 7 tens and 5 tenths as ‘705’, nobody could tell it from 7 hundreds and 5 ones. So we put a dot, the decimal point, just after the ones place. Everything to its left is the whole part; everything to its right is the fractional part.

Reading decimals. Read the whole part as a normal number, say ‘point’, then read the digits after the point one by one. We say 0.274 as ‘zero point two seven four’, not ‘zero point two hundred seventy-four’, because each digit has its own place.

How big? 1000 thousandths make 1 unit, 100 thousandths make 1 tenth, 10 thousandths make 1 hundredth, and 100 tenths make one ten.

Turning many small parts into a decimal. What is 234 tenths? 234/10 = 200/10 + 30/10 + 4/10 = 20 + 3 + 4/10 = 23.4. In the same way, 234 hundredths = 2.34 and 105 tenths = 10.5. Dividing by 10 moves every digit one place to the right of where it was.

Writing quantities. 2 ones, 3 tenths and 5 hundredths is 2.35. 4 ones and 6 hundredths is 4.06: the 0 in the tenths place is needed to keep the 6 in the hundredths place. 1 hundred, 1 one and 1 hundredth is 101.01.

Fractions with denominators 10, 100, 1000. 5/100 = 0.05, 16/1000 = 0.016, 12/10 = 1.2 and 254/1000 = 0.254. Some other fractions can be changed too: 1/2 = 5/10 = 0.5, 1/4 = 25/100 = 0.25, 3/4 = 0.75 and 1/5 = 2/10 = 0.2.

Most of the decimals we meet in daily life come from measurement: lengths, weights and money. Each pair of units is linked by 10, 100 or 1000, so decimals fit them perfectly.

Millimetres and centimetres. 1 cm = 10 mm, so 1 mm = 1/10 cm = 0.1 cm. Then 5 mm = 0.5 cm and 12 mm = 10 mm + 2 mm = 1.2 cm. Going the other way, 5.6 cm = 56 mm, because each centimetre has 10 mm.

Centimetres and metres. 1 m = 100 cm, so 1 cm = 1/100 m = 0.01 m. Then 15 cm = 15/100 m = 0.15 m, 4 cm = 0.04 m and 325 cm = 3.25 m. A metre has 1000 mm, so 1 mm = 0.001 m.

Grams and kilograms. 1 kg = 1000 g, so 1 g = 0.001 kg. Then 5 g = 0.005 kg, 68 g = 0.068 kg, 254 g = 0.254 kg and 1560 g = 1.56 kg. Also 1 g = 1000 mg, so 1 mg = 0.001 g.

Rupees and paise. 1 rupee = 100 paise, so 1 paisa = ₹0.01. Then 75 paise = ₹0.75, 5 paise = ₹0.05 and 250 paise = ₹2.50. Small coins of 25 paise and less were withdrawn in 2011, but paise still appear on bills and bank statements.

A quick rule. To change a small unit to a big one, divide by 10, 100 or 1000. To change a big unit to a small one, multiply. Always ask: should the number get bigger or smaller? 465 g is less than a kilogram, so the answer 0.465 kg must be less than 1.

Tiny things in nature. A human hair is about 0.1 mm thick, a sheet of newspaper is less than 0.1 mm thick, and some very small fish are less than 1 cm long. Decimals let us write such small sizes without inventing new units.

A big one. 1 km = 1000 m and 1 m = 1000 mm, so 1 km = 1000 × 1000 = 10,00,000 mm.

Where is 1.4 on a number line? It is 1 unit and 4 tenths. So cut the gap between 1 and 2 into 10 equal parts and count 4 parts from 1. The marks between 1 and 2 are 1.1, 1.2, …, 1.9.

Zooming in. To find 4.185, first find the gap from 4 to 5. Cut it into 10 parts and pick the gap from 4.1 to 4.2. Cut that into 10 parts and pick the gap from 4.18 to 4.19. Cut once more, and 4.185 is the middle mark. Each zoom shows one more decimal place.

Reading a number line with a different scale. If the marks between 5 and 10 split those 5 units into 10 parts, each part is half a unit, 0.5. Then the marks are 5.5, 6, 6.5, 7, 7.5 and so on. Always work out what one gap is worth before reading.

Zeros at the end. Is 0.2 the same as 0.20 and 0.200? Yes. All of them are 2 tenths; the extra zeros just say there are no hundredths or thousandths. But 0.02 and 0.002 are different: the 2 has moved to a smaller place. So 4.5 = 4.50 = 04.50, and 4.05 = 4.050, but 4.05 is less than 4.5.

Comparing place by place. Which is larger, 6.456 or 6.465? Line them up at the point and compare from the left.

We stop at the first place where the digits differ. The later digits cannot change the answer, because even 9 thousandths is less than 1 hundredth.

More digits does not mean bigger. 2.05 has more digits than 2.5, but 2.5 is larger (5 tenths beat 0 tenths). And 13.800 is larger than 3.81 because 13 ones beat 3 ones.

Closest decimal. Which of 0.9, 1.1, 1.01 and 1.11 is closest to 1? The distances are 0.1, 0.1, 0.01 and 0.11. So 1.01 is closest. Which of 3.56, 3.65 and 3.099 is closest to 4? The distances are 0.44, 0.35 and 0.901, so 3.65 is closest.

Ordering a list. Arrange 2.567, 2.675, 2.768, 2.499, 2.698 from largest to smallest: compare the tenths first (7, 6, 6, 5, 4), then the hundredths for the tie: 2.768, 2.698, 2.675, 2.567, 2.499.

Priya needs 2.7 m of cloth and Shylaja needs 3.5 m. Together they need 2.7 + 3.5 metres. This is the same sum as 2 7/10 + 3 5/10, which is 5 12/10 = 6 2/10. In decimal form we just write the numbers one under the other with the points in a line:

2.7 + 3.5: tenths 7 + 5 = 12, write 2 and carry 1; ones 2 + 3 + 1 = 6. Answer 6.2 m.

Subtraction works the same way. 3.5 − 2.7: we cannot take 7 tenths from 5 tenths, so we borrow 1 one, making 15 tenths. 15 − 7 = 8 tenths, and 2 − 2 = 0 ones. Answer 0.8 m. So Shylaja needs 0.8 m more.

The golden rule: line up the points. Then tenths sit under tenths and hundredths under hundredths. If one number has fewer decimal places, fill the gaps with zeros. For 17 − 0.05, write 17.00 − 0.05 = 16.95. For 0.934 + 0.6, write 0.934 + 0.600 = 1.534.

A longer example. 75.345 + 86.691. Thousandths 5 + 1 = 6. Hundredths 4 + 9 = 13, write 3 carry 1. Tenths 3 + 6 + 1 = 10, write 0 carry 1. Ones 5 + 6 + 1 = 12, write 2 carry 1. Tens 7 + 8 + 1 = 16. Answer 162.036. The carrying is exactly the same as for whole numbers.

Decimal sequences. In 4.4, 4.8, 5.2, 5.6, 6.0, … we add 0.4 each time, so the next terms are 6.4, 6.8, 7.2. In 12.45, 11.95, 11.45, … we subtract 0.5 each time, giving 10.95, 10.45, 9.95. Doing these in your head is good practice.

Estimate first. Before adding 25.936 + 8.202, look at the whole parts: 25 + 8 = 33. Each decimal part is less than 1, so the sum is more than 33 but less than 33 + 2 = 35. The real sum, 34.138, fits. If your answer falls outside this range, you have made a slip. For a difference such as 34.505 − 18.1, the answer must lie between 34 − 19 = 15 and 35 − 18 = 17; it is 16.405.

Word problem. Mahi buys 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicum and 0.05 kg ginger. In hundredths: 25 + 30 + 50 + 20 + 5 = 130 hundredths = 1.3 kg.

A misplaced decimal point can cost a lot. Reading a medicine dose of 0.05 mg as 0.5 mg gives ten times the dose. A city office once paid out a huge sum because amounts in cents were read as amounts in euros. A plane once ran short of fuel because pounds and kilograms were mixed up. Always check the unit and the place of the point.

Things that look like decimals but are not. Not every ‘point’ number is a decimal of tenths.

• Time. ‘4.5 hours after noon’ is not 4:05 or 4:50. An hour has 60 minutes, so 0.5 hour = 5 × 6 = 30 minutes. The time is 4:30 p.m.
• Feet and inches. 1 foot = 12 inches. A gap of 2 feet 5 inches is not 2.5 feet; 2.5 feet is 2 feet 6 inches. A door made 2.5 feet wide for a 2 ft 5 in gap will not fit.
• Cricket overs. 1 over = 6 balls. ‘5.5 overs’ means 5 overs and 5 balls, not five and a half overs. There is never ‘5.6 overs’.

Why tenths and not quarters? We could split a unit into 4 or 16 parts, and some rulers and old measures do this. But our way of writing numbers is built on 10: ten ones make a ten, ten tens make a hundred. Splitting into tenths, hundredths and thousandths lets the same digits and places continue to the right of the point, so we can add and compare decimals exactly as we do with whole numbers.

A short history. Fractions with denominators 10, 100 and 1000 appear in the works of ancient Indian astronomers and mathematicians, including Śrīdharācārya in the 8th century. Around 950 CE, the Arab mathematician al-Uqlīdisī described decimal fractions in a book on Indian arithmetic, marking the end of the whole part with a small stroke. Later writers tried colours and small raised numbers. In the 16th century John Napier and Christopher Clavius used a point, while François Viète used a comma. Many countries still use a comma, writing 1 000,5 for our 1,000.5.

Checking habits.

• Read each digit after the point aloud separately.
• Ask whether the answer is sensible: a pencil is not 15.2 m long.
• Write the unit with every answer.
• For time, overs and feet, remember the unit is not split into 10 parts.