The big picture
Twelve ideas. Seven are about fractions and their decimals. Five are about numbers no fraction reaches.
| Idea | Remember |
|---|---|
| One shape | Any rational number is an integer over a non-zero integer. |
| No next door | Halfway between two fractions is another fraction. |
| Stopping | Cut down first. Only 2s and 5s in the bottom number make a decimal stop. |
| Repeating | A remainder that comes back makes every digit after it come back. |
| The road back | Two lines with the same tail, taken away, leave a whole number. |
| Irrational | A number that is not an integer over an integer. The diagonal of a unit square. |
| The argument | Suppose p/q is in lowest terms and squares to 2. Then p and q are both even. That is a clash. |
| The spiral | Each hypotenuse is the next leg. The square goes up by 1. |
| The squeeze | Square both ends of a gap and compare with the target. |
| Two names | The decimal 0.999… with endless 9s is exactly 1. A positive decimal that stops has a second name in endless 9s. |
| The cyclic block | Times 1 to 6, 142857 gives the same six digits, shifted. |
| Reading a decimal | A decimal that stops or repeats is rational. A decimal that never ends and never repeats is irrational. |