NCERT Class 8 Mathematics Ganita Prakash Part 1 Chapter 02 Power Play PDF Download

Chapter 2: Power Play

2.1 Experiencing the Power Play ...

An Impossible Venture!

Take a sheet of paper, as large a sheet as you can find. Fold it once. Fold it again, and again.

How many times can you fold it over and over?

Estu says "I heard that a sheet of paper can't be folded more than 7 times".

Roxie replies "What if we use a thinner paper, like a newspaper or a tissue paper?"

Try it with different types of paper and see what happens.

Say you can fold a sheet of paper as many times as you wish. What would its thickness be after 30 folds? Make a guess.

Let us find out how thick a sheet of paper will be after 46 folds. Assume that the thickness of the sheet is 0.001 cm.

[Figure: Cartoon showing children discussing paper folding and reaching the Moon, See in your textbook]

The following table lists the thickness after each fold. Observe that the thickness doubles after each fold.

(We use the sign '\( \approx \)' to indicate 'approximately equal to'.)

After 10 folds, the thickness is just above 1 cm (1.024 cm).

After 17 folds, the thickness is about 131 cm (a little more than 4 feet).

Now, what do you think the thickness would be after 30 folds? 45 folds? Make a guess.

Fill the table below.

After 26 folds, the thickness is approximately 670 m. Burj Khalifa in Dubai, the tallest building in the world, is 830 m tall.

After 30 folds, the thickness of the paper is about 10.7 km, the typical height at which planes fly. The deepest point discovered in the oceans is the Mariana Trench, with a depth of 11 km.

[Figure: Image of Burj Khalifa and children, See in your textbook]

It might be hard to digest the fact that after just 46 folds, the thickness is more than 7,00,000 km. This is the power of multiplicative growth, also called exponential growth. Let us analyse the growth.

We have seen that the thickness doubles after every fold.

Notice the change in thickness after two folds. By how much does it increase?

After any 3 folds, the thickness increases 8 times (= \( 2 \times 2 \times 2 \)). Check if that is true. Similarly, from any point, the thickness after 10 folds increases by 1024 times (= 2 multiplied by itself 10 times), as shown in the table below.

Teacher's Note

When you see "the thickness increases by 1024 times", that means you multiply the previous thickness by 1024. This is different from "increases by 1024 cm", which would mean you add 1024 cm. The paper folding problem is multiplicative - each fold multiplies the thickness by 2, not adds a fixed amount.

2.2 Exponential Notation and Operations

The initial thickness of the paper was 0.001 cm.

Upon folding once, its thickness became 0.001 cm \( \times \) 2 = 0.002 cm.

Folding it twice, its thickness became -

0.001 cm \( \times \) 2 \( \times \) 2 = 0.004 cm, or 0.001 cm \( \times 2^2 \) = 0.004 cm (in shorthand).

Upon folding it thrice, its thickness became -

0.001 cm \( \times \) 2 \( \times \) 2 \( \times \) 2, or 0.001 cm \( \times 2^3 \) = 0.008 cm.

When folded four times, its thickness became -

0.001 cm \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2, or 0.001 cm \( \times 2^4 \) = 0.016 cm.

Similarly, the expression for the thickness of the paper when folded 7 times will be 0.001 cm \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2, or 0.001 cm \( \times 2^7 \) = 0.128 cm.

We have seen that square numbers can be expressed as \( n^2 \) and cube numbers as \( n^3 \).

\( n \times n = n^2 \) (read as 'n squared' or 'n raised to the power 2')

Teacher's Note

The small number written above and to the right, like the 2 in \( n^2 \), is called the exponent or power. It tells you how many times to multiply the base number by itself. For \( 2^7 \), you multiply 2 by itself 7 times: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \). Do not confuse \( 2^7 \) with \( 2 \times 7 \) - they are very different!

Key Points

• When a thickness or quantity doubles repeatedly, it grows exponentially - each time the exponent increases by 1, the value multiplies by 2.
• Exponential notation like \( 2^7 \) means \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \) - multiply the base by itself as many times as the exponent says.
• In the paper folding problem, after \( n \) folds, the thickness is \( 0.001 \text{ cm} \times 2^n \), showing how exponential growth can lead to enormous numbers from small starting values.

This is a preview of the first 3 pages. To get the complete chapter, click below.