Real numbers
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The document discusses the evolution of different types of numbers. It begins with counting numbers, then expands to whole numbers with the inclusion of zero, followed by integers which include negative numbers. Fractions are introduced when splitting a whole into parts. Rational numbers include all integers and fractions. Irrational numbers like the square root of two are introduced, and real numbers encompass rational and irrational quantities.
Real numbers
- 2. What is the simplest idea of a number? Something to count with!
- 3. The Counting Numbers Counting Numbers: {1, 2, 3, ...} And the "Counting Numbers" satisfied people for a long time...
- 4. The Whole Numbers if there is nothing to count, how can you count it? The idea of zero was not natural to early humans ... difference between "4" and "40" ? Without the zero they look the same!
- 5. The Natural Numbers • the "Counting Numbers": {1, 2, 3, ...} • or the "Whole Numbers": {0, 1, 2, 3, ...} they disagree on whether zero is "natural" or not.
- 6. Negative Numbers We can count forwards: 1, 2, 3, 4, .. ... but what if we count backwards: 3, 2, 1, 0, ... what happens next? If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers
- 7. Int egers Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
- 8. Fr ac t i ons If you have one orange and want to share it with someone, you need to cut it in half. You have just invented a new type of number!
- 9. Rational Numbers • all the integers • and all fractions.
- 10. But There Is More... If you draw a square (of size "1"), what is the distance across the diagonal? The answer is the square root of 2, which is 1.4142135623730950...(etc) .. and it is not a rational number
- 12. Real Num bers Real Numbers include: • the rational numbers, and • the irrational numbers