Numbers

1. Numbers

3. Numbers These beasts come in several different families. The simplest are the NATURAL NUMBERS, , made up of normal counting numbers, 1, 2, 3, �

4. Numbers These beasts come in several different families. The simplest are the NATURAL NUMBERS, , made up of normal counting numbers, 1, 2, 3, � and the special number discovered (invented?) by the Indian mathematicians in about the year 300

5. Numbers These beasts come in several different families. The simplest are the NATURAL NUMBERS, , made up of normal counting numbers, 1, 2, 3, � and the special number discovered (invented?) by the Indian mathematicians in about the year 300 ZERO !

6. Numbers These beasts come in several different families. The simplest are the NATURAL NUMBERS, , made up of normal counting numbers, 1, 2, 3, � and the special number discovered (invented?) by the Indian mathematicians in about the year 300 ZERO ! So now we have 0, 1, 2, 3, etc. but we are not yet quite complete.

7. If we count down from, say, three, we get

8. If we count down from, say, three, we get 3

9. If we count down from, say, three, we get 3 2

10. If we count down from, say, three, we get 3 2 1

11. If we count down from, say, three, we get 3 2 1 0

12. If we count down from, say, three, we get 3 2 1 0 Now what?

13. If we count down from, say, three, we get 3 2 1 0 Now what? �1

14. If we count down from, say, three, we get 3 2 1 0 Now what? �1 �2

15. If we count down from, say, three, we get 3 2 1 0 Now what? �1 �2 �3 etc

16. We often think of these numbers as arranged along a line:

17. We often think of these numbers as arranged along a line: � �3 �2 �1 0 +1 +2 +3�

18. We often think of these numbers as arranged along a line: � �3 �2 �1 0 +1 +2 +3� This line goes off as far as we like (to infinity) in either direction.

19. We often think of these numbers as arranged along a line: � �3 �2 �1 0 +1 +2 +3� This line goes off as far as we like (to infinity) in either direction. We call all the numbers on this line the INTEGERS

20. We often think of these numbers as arranged along a line: � �3 �2 �1 0 +1 +2 +3� This line goes off as far as we like (to infinity) in either direction. We call all the numbers on this line the INTEGERS and they are described by this fancy way of writing the letter z:

21. There are lots of numbers which occur in between the integers � all the fractions, e.g. 3/5, 31/12, 0.317, 2.483 etc. All the fractions and integers together are called Rational Numbers , ,

22. There are lots of numbers which occur in between the integers � all the fractions, e.g. 3/5, 31/12, 0.317, 2.483 etc. All the fractions and integers together are called Rational Numbers , , because they can all be written as ratios of whole numbers.

23. There are lots of numbers which occur in between the integers � all the fractions, e.g. 3/5, 31/12, 0.317, 2.483 etc. All the fractions and integers together are called Rational Numbers , , because they can all be written as ratios of whole numbers. Ratio is just an old word for fraction.

24. There are lots of numbers which occur in between the integers � all the fractions, e.g. 3/5, 31/12, 0.317, 2.483 etc. All the fractions and integers together are called Rational Numbers , , because they can all be written as ratios of whole numbers. Ratio is just an old word for fraction. There are other numbers which cannot be represented by a fraction (unless we use an infinite number of decimal places).

25. There are lots of numbers which occur in between the integers � all the fractions, e.g. 3/5, 31/12, 0.317, 2.483 etc. All the fractions and integers together are called Rational Numbers , , because they can all be written as ratios of whole numbers. Ratio is just an old word for fraction. There are other numbers which cannot be represented by a fraction (unless we use an infinite number of decimal places). These are called Irrational Numbers

26. There are lots of numbers which occur in between the integers � all the fractions, e.g. 3/5, 31/12, 0.317, 2.483 etc. All the fractions and integers together are called Rational Numbers , , because they can all be written as ratios of whole numbers. Ratio is just an old word for fraction. There are other numbers which cannot be represented by a fraction (unless we use an infinite number of decimal places). These are called Irrational Numbers and some you will be familiar with are

27. v2 = 1.4142�

28. v2 = 1.4142� v3 = 1.732�

29. v2 = 1.4142� v3 = 1.732� p = 3.14159�

30. v2 = 1.4142� v3 = 1.732� p = 3.14159� (This is special type of irrational, called a transcendental number)

31. v2 = 1.4142� v3 = 1.732� p = 3.14159� (This is special type of irrational, called a transcendental number) All of these groups:- naturals, rationals and irrationals, when added together make up the Real Numbers, .

32. There is one final class of numbers whose members are not all in the REAL( ) group and these are the COMPLEX NUMBERS ( )

33. There is one final class of numbers whose members are not all in the REAL group and these are the COMPLEX NUMBERS which include things like v�1 or the square root of any other negative number.

34. There is one final class of numbers whose members are not all in the REAL(?) group and these are the COMPLEX NUMBERS (?) which include things like v�1 or the square root of any other negative number. You may not like complex numbers to start with (I remember being very disbelieving when I first met them) but, like the real numbers, they are extremely useful in calculations.

35. There is one final class of numbers whose members are not all in the REAL(?) group and these are the COMPLEX NUMBERS (?) which include things like v�1 or the square root of any other negative number. You may not like complex numbers to start with (I remember being very disbelieving when I first met them) but, like the real numbers, they are extremely useful in calculations. E.g. We would probably have no mains electricity or certainly no electronic gadgets (cellphones, computers etc.) if people did not use complex numbers in their design.