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Every number you know is a point on one line. Some points on that line are not fractions at all.

Chapter 3 of Ganita Manjari is The World of Numbers.

What you will learn

• One shape. Each number so far is one integer over another.
• No next door. No fraction has a fraction right after it. Between two different fractions there is always another.
• Decimals that stop. Which fractions give them, and why.
• Decimals that repeat. A block of digits comes back again and again.
• The road back. Turn a repeating decimal into a fraction.
• A length with no fraction. The diagonal of a square.
• Proving it. One argument rules out every fraction.
• The spiral. Right triangles built one after another.
• Narrowing a root. Find two numbers it lies between.
• Two names. 0.999… with endless 9s is exactly 1.
• A block that turns round. 142857 times 1 to 6. The same six digits come back.
• Reading a decimal. A decimal that stops or repeats is rational. Rational means a fraction. A decimal that never ends and never repeats is irrational.

Need a quick recap?

These earlier ideas help. You may skip them if you know them.

• Squares and roots. The square root of 9 is 3. It is 3 because 3² = 9.
• The right triangle. Draw a square on each side. The two small squares add to the big one.
• Primes. A prime has only 1 and itself as factors. Every whole number above 1 is a product of primes.
• Fractions on a line. Each fraction is a point on the line.