Class 8 Mathematics Chapter 2: Power Play

Fold a sheet and see

Take the largest sheet of paper you can find and fold it. Fold it again, and again. Most people stop at about seven folds, because the paper gets stiff and fat. A thinner sheet, such as newspaper or tissue, lets you go a little further. The limit is physical, not arithmetical.

Suppose folding never stopped

Now suppose you could keep folding as long as you liked. How thick would the sheet be after thirty folds? Guess before reading on. Nearly everyone guesses far too low, and the reason is worth understanding. The thickness does not go up in steps of the same size.

Each fold doubles

Folding puts the paper on top of itself, so the thickness goes up two times. Start with a sheet 0.001 centimetres thick. One fold makes 0.002, two folds 0.004, three folds 0.008. The amount added gets bigger each time, because it is always equal to everything you already had.

The first centimetre

The early numbers look harmless. After seven folds the sheet is 0.128 centimetres, which is thinner than a coin. After ten folds it is 1.024 centimetres, so it has just passed a single centimetre. Ten doublings have produced barely more than a fingernail of thickness.

Then it starts to climb

After thirteen folds the stack is 8.192 centimetres. After sixteen it is 65.536 centimetres. After seventeen it stands at about 131 centimetres, a little over four feet. The same doubling that took ten folds to reach one centimetre now adds half a metre in a single step.

Past buildings

After twenty folds the stack is about 10.4 metres, taller than a house. After twenty-six it is roughly 670 metres. Dubai holds the tallest building standing anywhere, the Burj Khalifa, at 830 metres. By the twenty-seventh fold the paper has gone past even that.

Past aircraft and ocean

After thirty folds the thickness is about 10.7 kilometres, which is about the height at which passenger aircraft cruise. The deepest point known in the oceans, the Mariana Trench, is around 11 kilometres. Thirty folds of a thin sheet span the deepest water on Earth.

And then to the Moon

After forty-six folds the figure passes 7,00,000 kilometres, which is far enough to reach the Moon. The claim sounds like a mistake, and the first reaction is always that some zeros have been dropped. They have not. This is simply what doubling does when it is allowed to continue.

Why guessing fails

A guess assumes the steps are roughly equal, because that is what addition does. Here each step is as large as the whole stack so far. Growth of this kind is called multiplicative growth, or exponential growth, and it is the subject of this chapter.

Ten folds, 1,024 times

There is a neater way to see the speed. Going from no folds to ten folds takes the thickness from 0.001 centimetres to 1.024 centimetres. Divide one by the other and the answer is 1,024. Two multiplied by itself ten times is 1,024, so ten doublings always multiply by that.

And it does not matter where you start

Check the same thing further along. From fold ten to fold twenty the thickness goes from 1.024 centimetres to about 10.485 metres, and that ratio is 1,024 again. From fold twenty to fold thirty, the same. From thirty to forty, the same. The factor depends on the number of steps, not on where they begin.

A rough rule for estimating

Since 1,024 is close to a thousand, ten doublings multiply by about a thousand and twenty doublings by about a million. That is enough to check any claim about doubling without a calculator, and it explains the Moon answer in one line.

A pond full of lotuses

A magical pond holds one lotus, and the number of lotuses doubles every day. After thirty days the pond is completely covered. On which day was it half covered? The answer is day twenty-nine, because the step from half to full is one doubling.

Reading growth backwards

That answer surprises people, who expect halfway in area to mean halfway in time. It never does under doubling. Going backwards from the end, the pond was a quarter covered on day twenty-eight and an eighth on day twenty-seven. The last day does half the work.

Two ponds, two rates

A second pond triples each day instead of doubling. Damayanti puts a lotus in the doubling pond, and after four days there are sixteen. She moves them all to the tripling pond, where four more days multiply them by three four times over. The order of the two ponds does not change the final count.

Scenario:

A school of 1,200 pupils hears a rumour. On day one, one pupil knows it. Each pupil who knows it tells exactly one new pupil the next day, so the number who know it doubles daily. The head teacher wants to know when half the school will have heard it. A teacher guesses that it will take about half of however long the whole school takes. The science club asks a second question. They want to know how many more pupils hear it on the final day than on all the earlier days put together.

Analysis:

The rumour doubles, so the pond question applies exactly. Working backwards from the end is easier than working forwards, and the final-day count answers the second question in one line.

Title:

Race addition against doubling

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

Adding reaches 21 and doubling reaches 10,48,576. The two columns start together and end a million apart, which is the whole lesson of the chapter in one page.

After twenty-six folds the stack would stand about 670 metres tall, which is shorter than the Burj Khalifa at 830 metres.

After thirty folds the stack reaches roughly the height at which passenger aircraft fly, and the Mariana Trench is only a little deeper than that.

Ten folds, a thousand times — Every ten doublings multiply by 1,024, which is close enough to a thousand for a quick estimate.

A shorter way to write a long product

The folding story needed the thickness after seven folds, which meant multiplying by two seven times over. Writing all seven twos is slow and easy to miscount. We already have shorthand for two of these: n times n is n squared, and n times n times n is n cubed. The same idea goes further.

The notation and its parts

For any number n, we write n with a small raised count beside it to mean n multiplied by itself that many times. Five to the power four means five times five times five times five, which is 625. Here five is the base and four is the exponent, also called the power.

Reading it aloud

Five to the power four can be read as five raised to the power four, or as the fourth power of five. Powers of one base make a family: five to the one, five squared, five cubed, five to the fourth, and so on are all powers of five. The base stays, the exponent counts.

Exponents shorten a factorisation

Break 32,400 into primes and the list runs long: four twos, two fives and four threes. In exponential form that is two to the fourth, times five squared, times three to the fourth. The same information fits on one line and nothing has to be counted twice.

Letters work the same way

A product of letters shortens too. a times a times a times b times b is a cubed b squared. And a times a times b times b times b times b is a squared b to the fourth. Each letter keeps its own exponent, counting only its own appearances.

Adding is not multiplying

It is worth stopping on this. Four plus four plus four is three fours, which is twelve. Four times four times four is four cubed, which is sixty-four. The same number appearing three times gives twelve under addition and sixty-four under multiplication.

A negative base

The base may be negative. Minus four cubed is minus four times minus four times minus four, which is minus 64. But minus two to the fourth is 16, because the four negative factors pair off. Minus one to an odd power is minus one, and to an even power is one.

Zero as a base

Zero squared is zero, and zero to the fifth is zero as well. Any counting-number power of zero is zero, because the product always contains at least one zero. That is worth noting now, because the case of a zero EXPONENT is a different question and comes later.

The riddle of the shining stones

Three daughters each get three baskets. Each basket holds three keys, each key opens three rooms, each room has three tables, each table holds three necklaces, and each necklace carries three diamonds. The rooms number three to the fourth, which is 81, and the diamonds number three to the seventh.

Multiplying powers of one base

We had already reached three to the fourth, which is 81. To find three to the seventh, multiply that by three cubed, which is 27. So three to the fourth times three cubed is three to the seventh, and 81 times 27 is 2,187. The exponents simply added.

Why the exponents add

Write it out. Three to the fourth is a run of four threes, and three cubed is a run of three threes. Multiplying them joins the two runs into a single run of seven threes. So for any base, a run of a factors times a run of b factors is a run of a plus b factors.

The same law with letters

p to the fourth times p to the sixth joins a run of four p factors to a run of six. That is ten p factors in all, so the answer is p to the tenth. The law in words: a power times a power of the same base keeps the base and adds the exponents.

A power of a power

Four to the sixth can be grouped as two runs of three fours each, which is four cubed, squared. It can also be grouped as three runs of two fours, which is four squared, cubed. Both come to 4,096. Grouping a run in different ways gives the same total.

Why the exponents multiply there

Two to the tenth is ten twos. Grouped in pairs it is five groups of two squared, so it is two squared to the fifth. Grouped in fives it is two to the fifth, squared. In each case the two exponents multiply to ten, which is the law: a power raised to a power multiplies the exponents.

Two bases, one exponent

Damayanti moved her lotuses from the doubling pond to the tripling pond, four days in each. The count was two to the fourth times three to the fourth. Regroup the factors in pairs of two and three and you get six multiplied by itself four times, which is six to the fourth. One exponent over two bases multiplies the bases.

Counting arrangements with a power

Estu has four dresses and three caps, giving twelve outfits. A five-digit lock has ten choices for each digit, so the passwords number ten to the fifth, or 1,00,000. A six-slot lock using A to Z has twenty-six to the sixth. Counting choices is where powers earn their keep.

Scenario:

A railway cloakroom offers two kinds of locker. The first takes a four-digit code, where each slot holds any digit from 0 to 9. The second takes a three-slot code where each slot holds any letter from A to Z. A traveller wants the locker that is harder to guess. The clerk says the four-digit one must be safer because four slots beat three. A second traveller asks how much safer the letter locker becomes if one more slot is added to it.

Analysis:

Each slot contributes a factor equal to the number of choices it offers, so the count is a power. Comparing the two lockers means comparing ten to the fourth with twenty-six cubed, and the extra slot multiplies by twenty-six again.

Title:

Build a power line

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

Every step right multiplies by four and every step left divides by four. The line turns the laws of exponents into distances.

A six-slot lock using the letters A to Z has twenty-six to the power six passwords, which is over three hundred million.

The riddle of the shining stones hides a power: three daughters, three baskets, three keys, three rooms, three tables and three necklaces of three diamonds is three to the power seven.

Same base add, stacked multiply, same power share — One base in a product adds exponents. A power of a power multiplies them. One exponent over two bases multiplies the bases.

Halving a line, and watching the exponent

Take a line 16 units long, which is two to the fourth. Erase half of it and 8 units are left, which is two cubed. Halve again for 4 units, which is two squared. Each halving takes one factor of two away, and the exponent drops by one.

The quotient law

Written as a division, two to the fourth divided by two cubed is 2, which is two to the first. The exponents subtracted: four take away three is one. In general, dividing powers of one base keeps the base and subtracts the exponents. So two to the hundredth divided by two to the twenty-fifth is two to the seventy-fifth.

Why the base cannot be zero

The law comes with a condition: the base must not be zero. The reason is simple. If the base were zero, the division would be zero divided by zero, and division by zero has no answer at all. So every law in this topic quietly assumes a non-zero base.

Pushing the law to a zero exponent

So far the top exponent has been the larger. What if they are equal? Two to the fourth divided by two to the fourth is plainly 1, because any number divided by itself is 1. The law says the answer is two to the power four take away four, which is two to the power zero.

So a zero exponent is one

Both routes must agree, so two to the power zero is 1. The same argument works for any non-zero base, because x to the power a divided by x to the power a is 1 and is also x to the power zero. This is not a rule invented for convenience; it is forced by the law we already had.

Pushing it one step further

Now halve a line of 16 units five times rather than four. Four halvings take us to 1, and the fifth takes us to a half. The law says the answer is two to the power four take away five, which is two to the power minus one. So two to the power minus one is a half.

Negative exponents are fractions

Halve the same line ten times and the law gives two to the power minus six. Writing out the factors leaves one over sixty-four, which is one over two to the sixth. So a negative exponent means one over that power. Ten to the power minus three is one over a thousand.

And it works the other way

If a negative exponent flips a power into a fraction, then a fraction with a negative exponent flips back. One over ten to the power minus three works out to ten cubed. So seven squared is one over seven to the power minus two. The flip is a two-way street.

A line of powers

Lay the powers of four along a line: 65,536, then 16,384, 4,096, 1,024, 256, 64, 16, 4, then 1, then a quarter, then a sixteenth. Each step to the right divides by four and each step to the left multiplies by four. The 1 sits where the exponent is zero.

Reading ratios off the line

The line makes comparisons easy. Four to the seventh is 16,384 and four to the fifth is 1,024. Their ratio is four squared, which is 16, so the first is 16 times the second. Any two marks on the line differ by a power of the base, read off by subtracting exponents.

Place value is powers of ten

We have written expanded forms for years without naming the powers. The number 47,561 is four ten-thousands, seven thousands, five hundreds, six tens and one. In powers of ten that is four times ten to the fourth, plus seven times ten cubed, plus five times ten squared, plus six times ten, plus one times ten to the power zero.

And past the decimal point

The same idea continues to the right of the point, using negative exponents. The number 561.903 is five hundreds, six tens, one unit, then nine tenths, zero hundredths and three thousandths. Those last three are nine times ten to the minus one, zero times ten to the minus two, and three times ten to the minus three.

Why very large numbers need help

Here are three facts. The Sun is about 30,00,00,00,00,00,00,00,00,000 metres from the centre of our galaxy. Our galaxy holds about 1,00,00,00,00,000 stars. The Earth has a mass of about 59,76,00,00,00,00,00,00,00,00,00,000 kilograms. Reading those correctly is almost impossible, and miscounting the zeros changes everything.

Scientific notation

Any number can be written as a coefficient times a power of ten. 5,900 is 5.9 times ten cubed. 20,800 is 2.08 times ten to the fourth. 80,00,000 is 8 times ten to the sixth. When the coefficient is at least 1 and under 10, this is called scientific notation, or standard form.

The exponent matters more

Writing the population of Mumbai as two times ten to the seventh puts the important information in the exponent. Changing the 2 to a 3 raises the figure by half. Changing the 7 to an 8 multiplies it by ten. That is why the exponent is read first and the coefficient second.

How many digits to write

The digits in the coefficient say how well the figure is known. If a population is certain to the unit it can be written out in full. If it is known only to about a lakh and forty-two thousand, write 1.42 times ten to the fifth. If only to about a lakh and forty thousand, write 1.4 times ten to the fifth and claim no more.

Naming the powers, long before writing them

The notation is recent; the need for it is not. Indian mathematical writing names the powers of ten one by one, far past anything daily life asks for. A hundred kotis make an ayuta, a hundred ayutas a niyuta, a hundred niyutas a kankara, and the ladder climbs from there. The Lalitavistara takes it to the fifty-third power of ten, Mahaviracharya lists twenty-four such names in the Ganita-sara-sangraha, and a Jaina work goes as far as the ninety-sixth. Each name is one rung; an exponent is the whole ladder in two symbols, which is the saving this chapter is about.

Scenario:

A newspaper prints two figures on the same page. The population of a city is given as 1,42,395 and the population of a district as 2 times ten to the seventh. A reader complains that the two are written in different styles. The editor replies that the district figure is only known roughly, while the city figure came from a count. A sub-editor then asks whether the district figure would change much if the 2 were a 3, and whether it matters more if the 7 were an 8.

Analysis:

The two styles carry different claims about certainty. The sub-editor question is answered by comparing the effect of a change in the coefficient with a change in the exponent.

Title:

Shrink a huge number

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

The long figures shrink to a digit, a point and a power. Deciding how many digits to keep is a judgement about how well the figure is known, not about arithmetic.

The Sun sits about 1.4335 times ten to the twelfth metres from Saturn, and about 1.496 times ten to the eleventh metres from Earth, so the Earth distance is the smaller by a whole power of ten.

Writing the population of Mumbai as two times ten to the seventh makes the seven the important digit: changing it to eight multiplies the figure tenfold.

Indian mathematicians had names for powers of ten long before the notation existed. A first-century BCE Buddhist treatise, the Lalitavistara, carries names for the odd powers as far as ten to the fifty-third.

Divide, so subtract; cross zero, so flip — Division subtracts exponents. Carry on past zero and the exponent goes negative, which flips the power into a fraction.