1. Number Systems

2. 2 View Video Clip. Please go to http://www.math.harvard.edu/~knill/math movies/swf/smilla2.html to view a short clip from the 1997 movie “Smilla’s Sense of Snow”, starring Julia Ormond and Gabriel Byrne.

3. Smilla's Sense of Snow3 The only thing that makes me truly happy is mathematics. Snow, ice, and numbers. The number system is like human life. First you have the natural numbers. The ones that are whole and positive. Like the numbers of a small child. But human consciousness expands. The child discovers longing. Do you know the mathematical expression for longing? The negative numbers. The formalization of the feeling that you're missing something.

4. 4 Natural Numbers, N. The most familiar numbers are the natural numbers or counting numbers: one, two, three, and so on. In the base ten number system, in almost universal use today for arithmetic operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is N, also written.

5. 5 Whole Numbers, W. The Natural (or “Counting”) Numbers, together with zero, make up the set of Whole Numbers. Thus the Whole Numbers can be used to represent the number of elements in a set, zero giving the cardinality of the Null Set {}. W = {0, 1, 2, 3, 4, … }

6. 6 Integers, Z. Negative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by a negative sign (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called integers, Z.

7. 7 Rational Numbers, Q. A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. In the fraction m / n, m represents equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1 / 2 and 2 / 4 are equal, that is ½ = 2 / 4. If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7 / 1. The symbol for the rational numbers is Q (for quotient).

8. 8 Real Numbers, R. The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus 123.456 represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point". In the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when necessary to indicate that it is to be treated as a real number rather than as an integer. Negative real numbers are written with a preceding minus sign: -123.456

9. 9 Real Numbers, R (continued) Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1 / 2 and the real number 0.333... (forever repeating threes) can be written as 1 / 3. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is 3.14159265… Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include the square root of 2, that is, the positive number whose square is 2.

10. 10 Complex Numbers, C. Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from the question of whether a negative number can have a square root. This led to the invention of a new number: the square root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. In the expression a + bi, the real number a is called the real part and bi is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a subset of the complex numbers. The symbol for the complex numbers is C.

11. 11 Subsets NW Z Q R C

12. 12 Subsets The Complex Numbers comprise all the Real Numbers, as well as the Imaginary Numbers and those with both real and imaginary parts. The Real Numbers are made up of both Rational and Irrational Numbers. The Rational Numbers include all the Integers. The Integers include both positive and negative Whole Numbers. The Whole Numbers are made up of the Natural Numbers plus zero. N W Z Q R C

13. 13 References / Sources: http://www.math.harvard.edu/~knill/mathmo vies/swf/smilla2.html http://www.imdb.com/title/tt0120152/quotes http://en.wikipedia.org/wiki/Number