2303 Math Unit 2 Building the Set of Real Numbers A Calculator Free Zone Mathematics developed by necessity. People needed to know quantities, so civilizations developed numeration systems. People needed to be able to put quantities together and split them up, so mathematical operations were developed. We'll see that the set of real numbers also came to be out of necessity. But that's getting a little ahead. We have seen the set of counting numbers (natural numbers): {1, 2,3, 4,...} This is sometimes noted as . Suppose a farmer is keeping track of his cattle. He counts the cattle in one pen. He counts the cattle in another pen. Then he goes into his barn, and there are no cattle there. Is the set of counting numbers enough for this situation?
What's needed is the concept of 0 the absence of value or quantity. There is no zero in . So man created a new set of numbers the set of whole numbers. It adds exactly one number to the set of counting numbers: {0,1, 2,3, 4,...} .
Operations with 0 Addition: a + 0 = 0 + a = a Zero is called the identity element for addition over the set of whole numbers.
Subtraction: a 0 = a (nice!) but 0 a = -a. Subtraction does not have an identity element because of the need for an identity element to be "double-sided." Multiplication: 0 * a = a * 0 = 0 Division: 0 a = 0 a 0 is not defined!!
We've already seen a limitation to . Subtraction is not a binary operation on this set. Recall 4 6 gives a result that is not in . What is needed?
Our next set and it's an important one! The Set of Integers
= {..., -3, -2, -1, 0,1, 2,3,...}
Note: We sometimes call the set of counting numbers "the set of positive integers." The following examples should show you what you need to be able to do with the set of integers. Example 1: Plot these integers on the real number line: -5, 2, 0, -1, 6
_______________________________________________________
Example 2: Insert either > or < in the space provided to make a correct statement. A. -3 _____ -5 B. 2 ______ 0 C. 0 ______ -1 D. -7 _____-3 Addition of integers: You can use a number line to add. Example 3: Compute each. A. -3 + 7
B. -2 + (-3)
C. 4 + (-5)
D. 5 + 6
Example 4: Find the absolute value of each: A. -2
B. 18
Example 5: Which of these numbers has the larger absolute value? A. -21 or 17
B. -3 or 5
Here are some generalizations about adding two integers: If the signs of the numbers are the same, add absolute values and use the common sign for your answer. If the signs of the numbers are not the same, subtract the smaller absolute value from the larger one. Use the sign of the number that had the larger absolute value. Example 6: Add without using a number line: A. -5 + 14
B. -3 + (-8)
C. 15 + (-17)
For a number a, we define the additive inverse of a to be a so that a + (-a) = (-a) + a = 0. Example 7: State the additive inverse of each. A. -9
B. 8
Example 8: Compute 5 + (-5)
You can apply the same rules above to add more than two integers. Just work with two at a time. Example 9: A. -9 + 7 + (-4)
B. -2 + (-9) + (-14) + 25
We'll continue to be interested in the properties of the operation within the stated set. Closure: Is the set of integers closed under addition?
Identity element: Is there an identity element for addition in the set of integers?
Associativity: Is addition associative over the set of integers?
Commutativity: Is addition commutative over the set of integers?
Subtraction of integers We'll use the definition of subtraction to convert any subtraction problem into an addition problem:
a - b = a + (-b) Once converted, use the rules for addition. Example 10: Rewrite each of these as an addition problem. Then use the rules for addition of integers to find your answer. A. 3 7
B. -2 11
C. 6 3
D. 2 (-8)
E. -9 (-6)
F. -4 (-4)
G. -3 5 (-6)
Properties of Subtraction over the set of Integers: Closure
Associative
Commutative
Identity Element
Multiplication of Integers
You should be able to easily multiply two natural numbers together. Case 1: Positive * Positive Example 11: Compute: 6 * 8
You should also be able to multiply by zero. Example 12: 14 * 0
Now we'll look at multiplication of two negative integers and of two integers of different signs. Case 2: Positive * Negative Look for a pattern: 5 * 4 = 20 5 * 3 = 15 5 * 2 = 10 5*1=5 5*0=0 5 * (-1) = ____ 5 * (-2) = ____ Conclusion: (+)(-) = _____ Case 3: Negative * Positive 2 * 5 = 10 1*5=5 0*5=0 -1 * 5 = _____ -2 * 5 = _____ Conclusion: (-)(+) = _________ Case 4: Negative * Negative 3 * (-5) = -15 2 * (-5) = -10 1 * (-5) = -5 0 * (-5) = 0 -1 * (-5) = _____ -2 * (-5) = _____
Conclusion: (-)(-) = _____
General rules: The product of two numbers with the same sign will be positive. The product of two numbers of different signs will be negative. You can extend these general rules to products of more than two numbers by working with your numbers two at a time.
Example 13: Compute each: A. -6 * -2
B. 6 * -7
C. -2 * 5
D. -3 * -1 * -5 * -6
Properties: Closure
Identity element
Associative
Commutative
Division of Integers
Recall the relationship between multiplication and division:
a b = c if c*b = a. i.e., 10 2 = 5 if 5 * 2 = 10
Recall that we can also use fractional notation to indicate division. Because of the relationship between multiplication and division, the rules for computation are quite similar.
General rules: The quotient of two numbers with the same sign will be positive. The quotient of two numbers of different signs will be negative.
Example 14: Compute each: A. 15 (-3)
B. -20 (-4)
C. -30 6
D. 45 9
Properties: Closure
Identity Element
Commutative
Associative
The Distributive Property of Multiplication over Addition
a (b + c) = ab + ac
Example 15: Verify the distributive property in this example: 7(3 + 9)
Proofs Using the Properties of Multiplication and Addition
You'll need to be able to supply the reasons in some simple proofs using the commutative, associative and distributive properties, as well as the definitions of basic operations and basic arithmetic facts.
Example 16: Provide the reasons for each of the steps in this problem:
7 + 9 + 3 + 5 +1 7 + ( 9 + 3) + ( 5 + 1) 7 + (3 + 9) + (1 + 5)
( 7 + 3) + 9 + (1 + 5) ( 7 + 3) + (9 + 1) + 5
10 + 10 + 5
(10 + 10 ) + 5
20 + 5 25
Example 17: Provide the reasons for each step:
5 + 6( x + 8) + 5 x 5 + 6 x + 6 8 + 5x 5 + 6 x + 48 + 5 x 5 + ( 6 x + 48 ) + 5 x 5 + ( 48 + 6 x ) + 5 x 5 + 48 + (6 x + 5 x) 5 + 48 + x(6 + 5) 5 + 48 + x(11) 5 + 48 + 11x 53 + 11x
Example 18: Provide the reasons for each step:
18*7 (20 - 2) *7 (20 + (-2)) *7
( 20 *7 ) + ( (-2) *7 ) ( 20 *7 ) + ( -1(2* 7) ) ( 20 *7 ) + (-14)
140 + ( -14 ) (126 + 14) + (-14) 126 + (14 + (-14)) 126 + 0 126
Back to Building the Set of Real Numbers... Recall division was not a binary operation on the set of counting numbers. Why?
How can we take care of this problem?
The Rational Numbers
If a number can be expressed as the quotient of two integers (where the denominator is not 0), then that number is a rational number.
= the set of rational numbers.
If you can write a number as a fraction, it's rational! Example 19: Rational or not?
A. 6 B. 0 C.
-3 4 2 3
D. 1
E. -9 There are some numbers that are not rational, and we'll get to those soon. Here are some tasks that you need to be able to do. Recall this unit is a calculator free zone!
Reduce fractions to lowest terms
A fraction is said to be in lowest terms when the numerator and denominator of the fraction are relatively prime.
To reduce a fraction to lowest terms, find the greatest common divisor (GCD) of the numerator and the denominator, and then divide both by the GCD.
Example 20: Reduce each fraction to lowest terms:
A. 36 45
B.
49 63
C.
35 140
Convert a Positive Mixed Number to an Improper Fraction and Vice Versa
3 which consists of an integer and a fraction. An 4 17 improper fraction is a number such as where the numerator is greater than the 3 denominator. A mixed number is a number as such 2
Mixed Number to Improper Fraction: 1. Multiply the denominator of the fraction by the integer. 2. Add the result to the numerator of the fraction 3. Place the result as the numerator with the original denominator as the final denominator.
Example 21: Convert each mixed number to an improper fraction:
A. 3 4 5
B. 4
1 6
Improper Fraction to Mixed Number 1. Divide the numerator by the denominator. Note the quotient and the remainder. 2. The quotient in step 1 is the integer in the mixed number. The remainder is the numerator of the fraction. Use the denominator of the original fraction as the denominator of the fraction in your answer. Example 22: Convert each improper fraction to a mixed number.
A. 23 8
B.
118 7
To deal with negative improper fractions or mixed numbers, ignore the negative sign, convert, then make your answer negative. C. -11 2
Rational numbers can be expressed as decimal numbers. Every rational number will be either a repeating decimal number or a terminating decimal number. 2 2 = 0.4 is an example of a terminating decimal and = 0.333.... = 0.3 is an example 5 3 of a repeating decimal. e.g.,
Converting a Fraction to a Decimal Number Divide the numerator of the fraction by the denominator of the fraction. Example 22: Convert each fraction to a decimal number. State if the decimal number is terminating or repeating.
A. 4 5
B.
7 8
C.
1 6
D.
37 33
Converting a Terminal Decimal Number to a Fraction 1. Determine the number of decimal places in your number. Call that number n. 2. The denominator will be 10 raised to the n power. 3. Place the digits to the right of the decimal point in the numerator of your fraction. 4. Reduce your fraction to lowest terms, if appropriate. Example 23: Convert each to a fraction, reduced to lowest terms.
A. 0.85
B. 0.018
C. 1.375
Converting a Repeating Decimal Number into a Fraction
This is a bit more difficult. To do this, we'll set the number we're working equal to n, then multiply both sides of that equation by a power of 10. Our objective is to subtract two equations so that the "repeating part" of the fraction drops out.... Easiest to show this by example.
Example 24: Convert each repeating decimal to a fraction.
A. 0.3
B. 0.26
C. 4.135
Multiplication of fractions:
a c ac = , b 0, d 0 b d bd
Reduce answers to lowest terms, if appropriate. Example 25: Multiply each.
A. 5 5 6 8
B.
-2 -4 3 7
3 1 C. 2 4 4 8
Division of Fractions:
a c a d ad = = , b 0, c 0, d 0 b d b c bc
The number
d c is called the reciprocal of . c d
Example 26: Divide each.
A. 1 6 5 5
B.
3 2 4 5
C.
-3 -2 8 5
Addition and Subtraction of Fractions Case 1: Like Denominators
a b a+b a b a -b + = , c 0; - = ,c0 c c c c c c
Example 27: Add or subtract as indicated.
A. 1 3 + 8 8
B.
15 9 - 16 16
If your fractions have unlike denominators, then you will need to find the LCM of the denominators. You can use this Fundamental Law of Rational Numbers to rewrite your fractions.
a a c ac = = , b 0, c 0 b b c bc
Example 28: Add or subtract as indicated.
A. 1 2 + 2 3
B.
5 3 - 12 10
C.
2 7 + 45 75
You can always find a rational number in between two given rational numbers. The easiest way to do this is to find the one that is halfway between them. To find the rational number that is halfway between two rational numbers, add the two rational numbers together and the divide your sum by 2. Example 29: Find the rational number that is halfway between:
A. 1 2 and 5 5
B.
3 4 and 8 5
You can use fractions to help you solve some word problems. Example 30: Student assistants are assigned to proofread
1 1 1 , , and of a 600 page 3 4 5 manuscript. The author will proofread the rest. How many pages will the author proofread?
Example 31: A recipe calls for 4
each will you need if you: A. double the recipe?
1 1 cups of flour and 1 cups of sugar. How much of 2 3
B. only make half the recipe?
Properties of Rational Numbers Closure
Identity Element
Commutative
Associative
So then, we now have ways to deal with negative numbers and rational numbers. We can work with terminating decimals and repeating decimals. What more could we want????
Remember the Pythagorean Theorem? In right triangle ABC with right angle C, a 2 + b 2 = c 2 .
Example 32: In right triangle ABC with right angle C, find c if a = 3 and b = 4
Example 33: In right triangle ABC with right angle C, find c if a = b = 1.
There is no rational number c such that c 2 = 2 . Time to get creative again! The new set of numbers numbers that are not rational is called the set of irrational numbers. An irrational number is a real number that cannot be represented as a fraction, i.e., its decimal representation is a non-repeating, non-terminating decimal. Example: 1.23233233323333233333... is an irrational numbe...