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The World of Numbers

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

Whole numbers, negatives and mixed numbers are all points on one line.

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.

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This track

  1. 0 Start here
  2. 1 Every number so far is one integer over another
  3. 2 There is always another one in the gap
  4. 3 Which fractions stop, and what decides it
  5. 4 A block that repeats is the number written out
  6. 5 Turning a repeating decimal back into a fraction
  7. 6 Checkpoint: fractions and the decimals they make
  8. 7 A length you can draw and cannot write
  9. 8 The argument that no fraction can do it
  10. 9 A spiral built one right triangle at a time
  11. 10 Squeezing it between two fractions
  12. 11 Checkpoint: the numbers that are not fractions
  13. 12 One number can have two decimal names
  14. 13 A block that turns round: 142857
  15. 14 The decimal tells you if a number is rational
  16. 15 The big picture
  17. 16 Track quiz

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