Class 9 - Chapter 3: The World of Numbers

Solution:

A rational number has a terminating decimal if the prime factors of its denominator (in simplest form) are only 2 and/or 5.

Checking by division:

7/20 = 0.35

4/15 = 0.26666...

13/250 = 0.052

Answer:

• 7/20 → Terminating decimal
• 4/15 → Repeating decimal
• 13/250 → Terminating decimal

Solution:

1/13 = 0.076923076923...

The repeating block is:

076923

Now observe:

The digits repeat in a cyclic order.

Each decimal contains the same six digits:

076923

but starting from a different position.

Answer: The repeating block is 076923, and the decimals exhibit a cyclic pattern.

Explicit Fractions:

√81 = 9 = 9/1

0.33333... = 1/3

23.560185612239874790120

= 23560185612239874790120 / 1000000000000000000000

(a terminating decimal can always be written as a fraction).

Solution:

Let

x = 0.99999...

Multiply both sides by 10:

10x = 9.99999...

Subtract the first equation from the second:

10x - x = 9.99999... - 0.99999...

9x = 9

x = 1

But x was defined as 0.99999...

Therefore,

0.99999... = 1

Answer: 0.99999... and 1 represent the same real number.

Solution:

A cyclic number is a repeating block whose successive multiples are cyclic rearrangements of the same digits.

Examples include:

These numbers are examples where the repeating decimal exhibits cyclic behavior.

Answer: Examples include 7, 17, 19, 23 and several other special prime numbers whose reciprocals generate cyclic repeating decimals.