← The World of Numbers

Lesson 15 of 17

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Twelve ideas. Seven are about fractions and their decimals. Five are about numbers no fraction reaches.

The World of Numbers — twelve ideas on one screen
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.