SECTION 3.1: What is a Rational

Transcription

1 1 NUMBERS: Artifacts from all over the world tell us that every culture had counting, tallying, and symbolic representations of numbers. Our number system has just ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Developed by Hindus, India, between 300 and 500 A.D. (Aces Research, Inc, ).

2 2 NUMBER SYSTEMS: Natural Numbers are the positive whole numbers. They are the numbers we use for everyday counting. The symbol, N, represents the natural numbers. N = {1, 2, 3, } The concept of numbers evolved over time, starting with the counting numbers (now called the natural numbers), which describe real-world quantities such as amounts, distances, age, and number items.

3 3 If we include ZERO with the natural numbers, we call the result the set of whole numbers. Adding ZERO to the counting numbers allowed humans to develop a number system with PLACE VALUE, that is, the position of a number determines its value. Whole Numbers are the positive whole numbers, including zero. The symbol, W, represents the whole numbers. W = {0, 1, 2, 3, }

4 4 Integers - are all the positive and negative whole numbers, including zero (the number line). The symbol, I, represents the Integers. I = {, -3, -2, -1, 0, 1, 2, 3, } Note: Another symbol that is sometimes used for the set of integers is the letter Z. (Aces Research, Inc, ).

5 5 Rational Numbers are made of ratios, as the name suggests. These are numbers that can be written as a quotient of two integers, that is, in the form a/b, where a and b are integers and b 0. (Cannot divide by zero!) **Rational numbers include ALL integers, fractions, terminating decimals, and repeating decimals. The symbol, Q, represents the rational numbers. Examples: -2, 7, 3, 0, 1, 0.333,

6 6 Irrational numbers - are numbers that are not rational. They are numbers that cannot be written as a quotient of two integers. **These numbers are decimals that are non-terminating and non-repeating. The symbol, -Q, represents the irrational numbers. Examples: л = = =

7 7 REAL NUMBERS - include ALL the rational and irrational numbers. That is, real numbers are natural, whole, integers, rational and irrational. The symbol, R, represents the real numbers. Draw diagram here!!

8 8 BEYOND REAL NUMBERS: Real numbers can handle most of our everyday calculations. Some applications in science and technology go beyond real numbers. These applications use numbers such as imaginary numbers (ai) and complex numbers (a + bi).

9 9 MATHEMATICIANS Some of the GREAT mathematicians were: Al- Khwarizmi (Intro. of Arabic numerals) Argand (complex number) Bombelli, Rafael (symbolic algebra) Cardano, Girolamo Descartes (x-y co-ordinate plane) Fermat (Fermat s last theorem: x n + y n = z n, states that for n >2, the equation has no solution) Fibonacci (Fibonacci sequence:1, 1, 2, 3, 5, 8, 13, 21, 34, )

10 10 Eratosthenes (Measuring the Earth s circumference) Euclid (Father of Geometry) Gauss (general arithmetic sequence) Goldbach (Goldbach s conjecture every even number > 2 is the sum of two primes) Kepler (planets move elliptical around the sun) Pascal (theory of probability) Pythagoras (Pythagorean Theorem) (Aces Research, Inc, )

11 11 NAMING VERY LARGE AND SMALL NUMBERS Large Numbers Thousand 10 3 Million 10 6 Billion 10 9 Trillion Quadrillion Quintillion Sextillion 10 21

12 12 Septillion Octillion Nonillion Decillion Undecillion Duodecillion Tredecillion Quatturodecillion10 45

13 13 VERY SMALL NUMBERS Milli 10-3 Micro 10-6 Nano 10-9 Pico Femto Atto 10-18

14 14 FOCUS: Compare and order rational numbers. Example 1: Find 2 rational numbers between and Solution: There are many answers!

15 15 Example 2: Order the following rational numbers in ascending order: 0.25, 0.444, -3.4, 0.9, -0.3 Solution: See Board!! Example 3: Order the following rational numbers in ascending order , - 4 7, , 1 3 4, Solution: See Board!!

16 16 Example 4: Indicate with a check mark what set each number belongs to: 4-5 2/ π N W I Q - Q R

17 17 Solution: N W I Q -Q R 4-5 2/ π

18 18 Example 5: Place the following on the number line. -0.5, 3, ½, 3/5, -5/6, -2.7, 9 Solution: See board solution!!

19 19 Example 6: What set does each number belong to? 4, 2, 0.5, -1.4 Solution: 4 N, W, I, Q, R 2 -Q, R 0.5 Q, R -1.4 Q, R

20 20 HOMEWORK: Textbook: page 101: #5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, and 25. Extra Practice 1: #1 to #6. Worksheet: Fractions #3-1(1):#1 to #24.