Decimals, and dividing by tens

Decimals (in Hindi, दशमलव) are the natural extension of the Indian place value system, used to represent the decimal fractions 1/10, 1/100, 1/1000 and so on, and their sums.

So 27.53 names a quantity with 2 tens, 7 units, 5 tenths and 3 hundredths. Every digit sits in a place worth ten times the one to its right.

That makes it easy to move between fractions and decimals. Take 254/1000. Split the numerator: 200/1000 + 50/1000 + 4/1000. Rewrite each in its own place: 2/10 + 5/100 + 4/1000. As decimals that is 0.2 + 0.05 + 0.004 = 0.254.

Jonali buys spices — 50 g of cinnamon, 100 g of cumin, 25 g of cardamom and 250 g of pepper. Write each in kilograms as a fraction and as a decimal: 50/1000 kg = 0.05 kg, 100/1000 = 0.1 kg, 25/1000 = 0.025 kg, 250/1000 = 0.25 kg.

Notice what those divisions have in common: every denominator is 1 followed by zeroes. Is there a simple rule for dividing ANY number by 10, 100, 1000 and so on?

There is. Step 1: write the dividend as it is, with a decimal point at the end — for 123, write 123. Step 2: count the zeroes in the divisor — 10 has one zero. Step 3: move the decimal point LEFT by that many places, adding zeroes in front if needed. So 123 ÷ 10 = 12.3.

The rule handles everything: 24 ÷ 100 = 0.24, 678 ÷ 1000 = 0.678, 12 ÷ 1000 = 0.012 (two zeroes had to be added in front), and 12345 ÷ 1000 = 12.345. This chapter extends multiplication and division to decimals, and you will find both are natural extensions of what you already do with counting numbers.