Representing quantities, relations and change
Ten marks still represent ten units; organization changes, not quantity.
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The ideas to retain
1. We invented languages to describe the world
Before writing, we were already counting
The need to count is far older than writing. Paleolithic notched objects such as the Lebombo bone and the Ishango bone do not by themselves prove the existence of formal mathematics, but…
Numbers, position and zero
For millennia, different civilizations developed their own ways of representing quantities. The Egyptians used an additive system, as did the Romans. The Babylonians introduced a powerful…
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Grouping does not change the object count
Start with seven units and add three more.
Regrouping them into two groups of five makes the total easier to check.
Ten marks still represent ten units; organization changes, not quantity.
Zero preserves a position
In one hundred five, one occupies the hundreds column and five the units column.
Zero indicates that the tens column is empty.
Removing that zero gives fifteen; moving a digit changes its place value.
The same operation preserves equality
The equation x plus three equals eight contains an unknown quantity.
Subtract three from both sides, not just one.
This leaves x equals five; the symbol lets us operate before knowing its value.
A compact rule describes many terms
Adding one, two, three, four, five and six gives twenty-one.
Sigma notation expresses the same sum with an index and two limits.
The notation compresses the description; it does not automatically perform the computation.
A conclusion depends on its premises
Assume every element of A also belongs to B.
If x belongs to A, those two premises imply that x belongs to B.
Deduction preserves the logical relation; it does not itself prove the premises describe the world.
Measuring change over smaller intervals
For y equals x squared, compare x equals one with one plus h.
The average slope is two plus h; as h shrinks it approaches two.
The limit describes local change, not an average over a large interval.
