Math-2A Lesson 2-1. Number Systems
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1 Math-A Lesson -1 Number Systems
2 Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure
3 Why do we need numbers? Lebombo Plain (Africa) Lebombo counting sticks appeared about 35,000 years ago! How can you write the number zero using a counting stick? How can you write a negative number using a counting stick?
4 A few horses? How do you count?
5 Vocabulary Natural numbers: the positive counting numbers that are usually shown on a number line Whole numbers: the natural numbers and the number zero Integers: the whole numbers and the negative counting numbers
6 Can anyone interpret what the following means? Rational numbers R : R a ; a,b integers b Vocabulary Rational numbers: can be written as a ratio of integers: ½, -⅔, etc.
7 When converting a rational number into its decimal form (using division) the decimal with either terminate (1/ = 0.5) or repeat (/3 = ). Write the integer -3 as a rational number Are these all the,,, same thing? Why is -3 not equal to?
8 If the triangle below is a right triangle, how can we find length c (the hypotenuse)? Pythagorean Theorem: If it s a right triangle, then side lengths can be related by: a b c 1 c 5 c 5 c What numbers system does this number belong to?
9 Property of Equality: if the same operation is applied to both sides of an equal sign, then the resulting equation is still true (has the same solution). x 4 x 4 Square both sides of the equation x 4 The same value of x makes both the 1 st and last equation true (x = ). x
10 Vocabulary Irrational numbers: cannot be written as a ratio of integers: ½, -⅔, etc. The decimal version of an irrational number never terminates and never repeats. (0 = ). If we see the radical symbol, the number is usually irrational (unless it is a perfect square). 3 4 (rational #)
11 3 Identifying the type of number. (1) () (3) (4) (5) Natural Whole Integer Rational Irrational
12 Exact vs. Approximate: Exact: 17 Approximate: Converting an irrational number into a decimal requires you to round off the decimal somewhere.
13 Irrational Numbers The square root of x really means, what number squared equals. x Why do these both refer to the same number (that makes both equations true)? Because of the Property of Equality.
14 1 The square root of -1: x 1 really means, what number squared equals -1. x 1 What real number when squared becomes a negative number? It doesn t exist so it must be an imaginary number 3 (1)* 3 (1) * 3 i 3
15 Vocabulary imaginary numbers: a number that includes the square root of a negative number. 1 i 3 3 real numbers: a number that can be found on the number line
16 Think of the complex numbers as the universe of numbers. Complex Numbers a b Real # s a b Imaginary # s
17 Complex Numbers Venn Diagram Imaginary Numbers 1 Integers Real Numbers Rational Numbers Irrational Numbers e i,-, -1, 0, 1,, 3, 4, i
18 Complex # s Real # s Imaginary # s Rational # s Irrational # s Integers Whole # s Natural # s 1 3 natural natural natural Is this always true? 3* 6 natural* natural natural Is this always true? 1 1 integer integer integer Is this always true? natural - natural natural Is this always true?
19 Which number system came first? Man probably invented the natural number system first. When the idea of zero was no longer scary, then it was probably added to form the whole number system. Then smart people started doing math with the numbers in the system. 1 What s wrong with this? 1? 3 duh! One subtract two is not in the whole number system!!! They needed a new number system!!!
20 Vocabulary Closure: a number system is closed for a particular operation (add, subtract, multiply, divide, etc.) when two numbers have an operation performed on them and the resulting number is still in the number system. We say that whole numbers and natural numbers are not closed under subtraction. Is there another operation for which the whole numbers or the natural numbers are not closed? 7 0? 1?
21 Associative Property of (multiplication) If there are 3 or more factors, you must pick two of them to combine first. *3*4 (*3)* 4 6* 4 4 Associative Property of (Addition) If there are 3 or more addends, you must pick two of them to combine with addition ( 3) 4 5 4
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