Math-2 Section 1-1. Number Systems
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1 Math- Section 1-1 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 a : R ; a,bintegers 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 Mathematical Property 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 4 x 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 Good to here
18 1 3 Complex # s natural natural natural Rational # s Integers Whole # s Natural # s Real # s Irrational # s Imaginary # s 1 1 integer natural natural Is this always true? 1 1 integer natural integer Is this always true? 1 1 integer natural rational Knowing how to combine numbers is important when simplifying expressions.
19 Complex Numbers Venn Diagram Imaginary Numbers 1 Integers Real Numbers Rational Numbers Irrational Numbers e i,-, -1, 0, 1,, 3, 4, i
20 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!!!
21 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?
22 New number systems are needed when a number system is not closed for a particular operation (the square root of -1) What number system is closed for all operations? The Complex Number System. a + bi + 3i Real number Imaginary number
23 Adding and Subtracting Complex # s ( + 3i ) + ( 4 + 7i ) =? Real numbers are NOT like terms with imaginary numbers. ( + 4 ) + ( )i
24 ( 3i) (-4 5i) =? Your Turn: 6 + i 7i ( 3i) =? i a 3i = 4 + bi a =?, b =? a = 4, b = -3
25 Multiplying Complex Numbers 3i * 4i = 3 * i * 4 * i = 3 * 4 * i * i 1i 1 i 1
26 Multiplying Complex Numbers (4 + 3i) = 8 + 6i (4 + i)(3 + 5i) = 4(3 + 5i) + i(3 + 5i) = 1 + 0i + 6i + 10i² = 1 + 6i + 10(-1) = + 6i
27 Additional material 1. The reason why we want to use i instead of is because mathematical operations are much easier for letters than with 1. Multiplication is repeated addition. x + x + x = 3x 3. Exponents are repeated multiplication. 1 x used as an addend 3 times is the same as 3 times x. x* x* x* x x x used as a factor 4 times is the same as x with an exponent of 4. 4
28 Additional material 4. If we combine items 1,, and 3 on the previous slide we have: 3 i i * i ( 1) * i 5. touching means multiplication. x *3x * x*3* 6. Commutative Property (of multiplication or addition): the order of the addends doesn t matter. x i 3 3 the order of the factors doesn t matter *3 3* You can rearrange the order if it makes it easier.
29 Additional material 7. Putting 5 and 6 from the previous slide together, we can do the following: x*3x * x*3* x *3* x * x 8. We an only multiply (or add) a pair of numbers in one step. *3* 4 (*3)* 4 6* Combining 3, 7, and 8 we have x*3x * x*3* x *3* x* x 6x
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Math 8A. Content Description Content Location U01-L01-A05. Learn: Text. Video U04-L18-A05. Learn: Text and. Video. Learn: Text and U04-L19-A03.
Know that there are numbers that are not rational, and approximate them by rational numbers. NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number
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Glossary. Glossary 981. Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle
1. Write three things you already know about expressions. Share your work with a classmate. Did your classmate understand what you wrote?
LESSON 1: RATIONAL EXPONENTS 1. Write three things you already know about epressions. Share your work with a classmate. Did your classmate understand what you wrote?. Write your wonderings about working
Ch. 7.6 Squares, Squaring & Parabolas
Ch. 7.6 Squares, Squaring & Parabolas Learning Intentions: Learn about the squaring & square root function. Graph parabolas. Compare the squaring function with other functions. Relate the squaring function
CK-12 Middle School Math Grade 8
CK-12 Middle School Math aligned with COMMON CORE MATH STATE STANDARDS INITIATIVE Middle School Standards for Math Content Common Core Math Standards for CK-12 Middle School Math The Number System (8.NS)