NUMBER SYSTEMS Section 1 3 Irrational numbers C

NUMBER SYSTEMS Section 1 -3 Irrational numbers C. N. Colón Algebra St. Barnabas H. S. Bronx, N. Y.

• There are infinitely many real numbers that can be written as decimals which neither terminate nor repeat. • These numbers are known as irrational numbers. • Think of an irrational number as a number that does not behave when expressed as a decimal, such as a square root that is not perfect.

Because of its lack of behavior, we can come to a good definition for an irrational number An irrational number is an infinite non-repeating decimal. It CANNOT be expressed as , where a and b are integers and b does not equal zero.

LET’S REVIEW A FEW TERMS To square a number means to multiply a number by itself. The square of 6 is 36 Calculators have a special key to square a number.

LET’S REVIEW A FEW TERMS The opposite of squaring is to find the square root. To find the square root is to find the number, that when multiplied by itself will give the value under the radical. The square root of 25 equals 5 because 5 ·5 = 25

LET’S REVIEW A FEW TERMS Your calculator also has a special key to find the square root It is the same key but you have to press the yellow 2 nd function key FIRST!

The best value that can be given for an irrational number is a rational approximation. = 3. 141592654… So we say that on and on 3. 14

ROUNDING • Using a calculator is very helpful. • Look at the digit to the right of the decimal place that you are rounding off to. • If that digit is less that 5, then the number in the place value you are rounding to stays the same and you can drop all the numbers to the right of it. • If that digit is equal to or greater than 5, then the number in the place you are rounding to increases by one and you can also drop all the numbers to the right of it.

ASSIGNMENT p. 23 # 4 -52, multiples of 4 (mof)

NUMBER SYSTEMS Section 1 -4 The Real Numbers C. N. Colón Algebra St. Barnabas H. S. Bronx, N. Y.

All real numbers are either rational or irrational. and can be ordered or compared in one of two ways: 1. They can be placed on the number line 2. They can be expressed as decimals and you can see which is greater.

Remember! The hungry alligator always eats the largest number.

Vocabulary Terms Throughout this course we will be working with real numbers. Real numbers can be pictured as points on a horizontal line called the real number line The point labeled 0 is called the origin Points to the left of zero represent negative numbers Points to the right of zero represent positive numbers Zero is neither positive nor negative.

Real Number Line Negative Numbers -5 -4 -3 -2 -1 Positive Numbers 0 1 Origin 2 3 4 5

Vocabulary Terms In the real number line the scale marks are equally spaced and represent integers. Integers are the set of numbers that include positive numbers, negative numbers and zero The real number line has points that represent fractions and decimals as well as integers. The point that corresponds to a number is called the graph of the number Drawing the point is called graphing or plotting the number

Graphing Real Numbers To graph a real number: 1. Draw and label a number line 2. Find where the number is on the number line and place a dot on the number Let’s look at an example…

Example # 1 Graph the numbers ½ and -2. 3 on a number line 1. Draw and label a number line -5 -4 - 2. 3 -3 -2 -1 0 ½ 1 2 3 4 5 2. Find where the number is on the number line and place a dot on the number In this instance ½ is positive so it is to the right of zero and less than 1 -2. 3 is negative and to the left of zero 2. 3 units

Comparing Real Numbers Once points have been plotted on the number line you can compare the numbers. Let’s look at an example N. B. : Be careful there is more than 1 way to read the comparison

Example # 2 Graph – 4 and – 5 on a number line. Then write 2 inequalities that compare the numbers -5 -4 -3 -2 -1 1. Plot the point – 4 0 1 2 3 4 5 2. Plot the point – 5 On the graph – 5 is to the left of – 4 , so – 5 is less than – 4 which is written as – 5<– 4 On the graph – 4 is to the right of – 5 , so – 4 is greater than – 5 which is written as – 4 > – 5

Comments About a number line…. it is expected that you already know the following: The further to the left of zero the smaller the number is…. The further to the right of zero the larger the number is….

Ordering Real Numbers Often times you will be asked to order numbers from least to greatest or greatest to least. The goal here is to see if you understand the relationship of the numbers…that is… do you know which number is bigger or smaller than another number You can use the number line help you by plotting the numbers and then reordering them… Another strategy you can use here is to convert the numbers to decimals and then plot and order them as requested Let’s look at an example…

Example 3 The solution is: -2, -3/2, 0, ½ , 1. 5, 4 List in increasing order: - 2, 4 , 0, 1. 5, ½ , - 3/2 • First convert ½ and -3/2 to decimals. Which are. 5 and – 1. 5 respectively Then plot each number on the number line: -2 -5 -4 -3 0 1. 5 -1. 5 - 3/2 -2 -1 ½ 0 1 1. 5 4 2 3 4 5 Since you are numbering in increasing order read the number line from left to right.

Comments In the previous example we converted 2 fractions into decimals to make it easier to plot on the number line… When writing your final solution to the problem you should always use the original numbers. In this case it was ½ and – 3/2 which are displayed in the solution

Opposites In a number line two numbers that are the same distance from zero but in opposite directions are called opposites… Let’s look at an example…

Example # 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 In the above number line 2 and – 2 are the same distance from zero but on opposite sides. Therefore they are considered opposites

Absolute Value The absolute value of a number is simply the distance the number is from zero on a number line. Vertical bars | | are used to represent the absolute value of a number The symbol |a| is read as “the absolute value of a”. The next slide shows a summary of absolute value

Absolute Value Example If a is a positive number then the |a| = a |3| = 3 If a is zero, then |a| = 0 |0| = 0 If a is a negative number, then the | - a| = a |- 3| = 3 Notice: The absolute value of a number is NEVER negative…I like to say that the absolute value of a number is always positive!

Comments On the next couple of slides are some practice problems… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn Graph the numbers on a number line then write 2 inequalities that compare the numbers -6 and 4 2. -6. 4 and -6. 3 3. -7 and 2 4. -2. 7 and 3/4 1.

Your Turn Draw a number line, plot the points and then write the numbers in increasing order 4. 8, – 2. 6, 0, -3, ½, - ½ 6. 7, - ½ , 2. 4, - ¾, - 5. 8, 1/3 7. 3 ½ , 3. 4, 4. 1, -5. 1, -4 ½ 5.

Your Turn Find the opposite of the number 8. 10 9. -3 10. 3. 8 11. -2. 5

Your Turn Find the absolute value |7| 13. |-4| 14. | ½ | 15. |0| + 2 12.

Your Turn Solutions 1. 2. 3. 4. 5. 6. 7. -6 < 4 or 4 > -6 -6. 4 < -6. 3 or -6. 3 > -6. 4 -7 < 2 or 2 > -7 -2. 7 < ¾ or ¾ > -2. 7 -3, -2. 6, - ½ , 0, ½ , 4. 8 -5. 8, - ¾ , - ½ , 1/3, 2. 4, 7 -5. 1, -5, -4 ½ , 3. 4, 3 ½ , 4. 1 8. 9. 10. 11. 12. 13. 14. 15. -10 3 -3. 8 2. 5 7 4 ½ 2

Summary A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about graphing, ordering and comparing numbers on the real number line as well as opposites and absolute value… Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you points for doing this lesson…please see the next slide…

BONUS POINTS I will add 25 points as an assignment grade for your work on this lesson…To receive the full 25 points you must do the following: Have a proper heading on the looseleaf Do the “your turn” problems showing all work Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted if any of the above is missing. Just because you hand in “something” does not mean you will get credit.

Mc. Donald’s Menu…. I’m Lovin It!! Double Cheeseburger: $. 99 Big Mac Value Meal: $ 4. 79 Chicken Mc. Nuggetts Meal: $ 3. 80 Small Drink: $. 99 Mc. Flurry: $ 1. 97 Salad: $ 4. 80 2 Cheeseburger Meal: $ 3. 70 Ice Cream Cone: $. 87

Order Up! Least Expensive to Most Expensive Ice Cream Cone Double Cheeseburger: Small Soft Drink: Mc. Flurry: 2 Cheeseburger Meal: Chicken Mc. Nuggets Meal: Big Mac Value Meal: Chicken Salad: . 87. 99 1. 97 3. 70 3. 80 4. 79 4. 80

What Do I Mean Compare Decimals? When we compare we use terms such as: Less than < Greater than > Equal to = Comparing decimals is similar to comparing whole numbers. 45<47 150>105 When we compare decimals we use place value or a number line.

Place Value 1, 000 10 1 0. 01 0. 0001

Compare Sara’s score with Half Pipe Results Danny’s score. 1. Line Up Decimal Points Danny 42. 1 Sara: 42. 5 Danny: 42. 1 Sara 42. 5 Ross 42. 0 Bethany 40. 7 Jacob 46. 1 2. Start at the left and find the first place where the digits differ. Compare the digits 1<5 This means Danny’s score was less than Sara’s score or that Sara’s score was greater than Danny’s 42. 5 > 42. 1 or 42. 1 < 42. 5

Let’s Try Using A Number Line Danny 42. 1 Sara 42. 5 Ross 42. 0 Bethany 40. 7 Jacob 46. 1 42. 0 42. 1 42. 5 Numbers to the right are greater than numbers to the left. Since 42. 5 is to the right of 42. 1 we have: 42. 5 > 42. 1

Equivalent Decimals 0. 60 and 0. 6 Are these the same? ? ? Decimals that name the same number are called equivalent decimals.

0. 60 = 0. 6

Annexing Zeros This means placing a zero to the right of the last digit in a decimal. n 0. 60 n Although we added a zero, the value of the decimal did not change!! n Annexing or adding zeros is useful when ordering a group of decimals. n

Ordering Decimals We can order decimals from least to greatest or we can order from greatest to least. Let’s try an example: Order 15, 14. 95, 15. 8, 15. 01 from least to greatest

15, 14. 95, 15. 8, 15. 01 First, line up the decimal points 15 14. 95 15. 8 15. 01

15, 14. 95, 15. 8, 15. 01 Next, annex zeros so that each number has the same number of decimal places 15. 00 14. 95 15. 80 15. 01

15, 14. 95, 15. 8, 15. 01 Finally, use place value to compare the decimals. Always start from the left!! 15. 00 14. 95 15. 80 15. 01 ANSWER: 14. 95, 15. 01, 15. 8 n

One More Example…. exit ticket Order these numbers from greatest to least showing all work to line up, annex zeros, and list your answer. 35. 06, 35. 7, 35. 5, 35. 849

ASSIGNMENT p. 27 #4 -32 (mo 4) Chapter Review: p. 35 #1 -40 (all)