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Integers and the Number Line Natural Numbers 1, 2, 3, 4, 5, ... Natural Numbers 1, 2, 3, 4, 5, ... Natural Numbers 1, 2, 3, 4, 5, ... Whole Numbers 0, 1, 2, 3, 4, 5, ... Natural Numbers 1, 2, 3, 4, 5, ... Whole Numbers 0, 1, 2, 3, 4, 5, ... For the number 1 there will be an opposite number For the number 1 there will be an opposite number For the number 1 there will be an opposite number -1. For the number 1 there will be an opposite number -1. For the number 2 there will be an opposite number For the number 1 there will be an opposite number -1. For the number 2 there will be an opposite number For the number 1 there will be an opposite number -1. For the number 2 there will be an opposite number -2. For the number 1 there will be an opposite number -1. For the number 2 there will be an opposite number -2. For the number 3 there will be an opposite number For the number 1 there will be an opposite number -1. For the number 2 there will be an opposite number -2. For the number 3 there will be an opposite number For the number 1 there will be an opposite number -1. For the number 2 there will be an opposite number -2. For the number 3 there will be an opposite number -3. Natural Numbers 1, 2, 3, 4, 5, 6, ... Whole numbers 0, 1, 2, 3, 4, 5, 6, ... Natural Numbers 1, 2, 3, 4, 5, 6, ... Whole numbers 0, 1, 2, 3, 4, 5, 6, ... Integers ... -4, -3, -2, -1, 0, 1, 2, 3, 4, ... EXAMPLE Tell which integer corresponds to this situation: The temperature is 4 degrees below zero. EXAMPLE Tell which integer corresponds to this situation: The temperature is 4 degrees below zero. The integer -4 corresponds to the situation. EXAMPLE Tell which integer corresponds to this situation: The temperature is 4 degrees below zero. The integer -4 corresponds to the situation. The temperature is -4 EXAMPLE Tell which integer corresponds to this situation: A contestant missed a $600 question on the television game show “Jeopardy.” EXAMPLE Tell which integer corresponds to this situation: A contestant missed a $600 question on the television game show “Jeopardy.” EXAMPLE Tell which integer corresponds to this situation: A contestant missed a $600 question on the television game show “Jeopardy.” EXAMPLE Tell which integer corresponds to this situation: A contestant missed a $600 question on the television game show “Jeopardy.” EXAMPLE Tell which integer corresponds to this situation: The shores of California’s largest lake, the Salton Sea are 227 feet below sea level. EXAMPLE Tell which integer corresponds to this situation: The shores of California’s largest lake, the Salton Sea are 227 feet below sea level. EXAMPLE Tell which integer corresponds to this situation: The shores of California’s largest lake, the Salton Sea are 227 feet below sea level. EXAMPLE Tell which integer corresponds to this situation: The shores of California’s largest lake, the Salton Sea are 227 feet below sea level. The integer -227 corresponds to the situation. The elevation is -227 feet. Example 4 Stock Price Change. Tell which integers correspond to this situation: Hal owns a stock whose price decreased $16 per share over a recent period. Example 4 Stock Price Change. Tell which integers correspond to this situation: Hal owns a stock whose price decreased $16 per share over a recent period. The integer – 16 corresponds to the decrease in the stock value. Example 4 Stock Price Change. Tell which integers correspond to this situation: Hal owns another stock whose price increases $2 per share over the same period. Example 4 Stock Price Change. Tell which integers correspond to this situation: Hal owns another stock whose price increases $2 per share over the same period. The integer 2 corresponds to the increase in the stock value. We created the set of integers by obtaining a negative number for each natural number and also including 0. -5 0 5 To create a larger number system, called the set of rational numbers, we consider quotients of integers with nonzero divisors. The following are some examples of rational numbers: The following are some examples of rational numbers: 2 , 3 The following are some examples of rational numbers: 2 , 3 2 , 3 The following are some examples of rational numbers: 2 , 3 2 , 3 7 , 1 The following are some examples of rational numbers: 2 , 3 2 , 3 7 , 3, 1 The following 2 , 2 , 7 , 3, 3 3 1 are some examples of rational 0, numbers: The following 2 , 2 , 7 , 3, 3 3 1 are some examples of rational 23 , 0, numbers: 8 The following 2 , 2 , 7 , 3, 3 3 1 are some examples of rational 23 , 2.4, 0, numbers: 8 The following 2 , 2 , 7 , 3, 3 3 1 are some examples of rational 23 , 2.4, 0.17, 0, numbers: 8 The following 2 , 2 , 7 , 3, 3 3 1 are some examples of rational 23 1 , 2.4, 0.17, 10 0, numbers: 8 2 The number 2 3 The number 2 3 (read “negative two-thirds”) can also be named The number 2 3 (read “negative two-thirds”) can also be named 2 3 The number 2 3 (read “negative two-thirds”) can also be named 2 or 3 The number 2 3 (read “negative two-thirds”) can also be named 2 2 or 3 3 That is That is a b That is a a b b That is a a a b b b Evaluate a b Evaluate a b for a 35 Evaluate a b for a 35 and b 7 Evaluate a b for a 35 and b 7 Evaluate a b 35 for a 35 and b 7 Evaluate a b 35 7 for a 35 and b 7 Evaluate a b 35 7 for a 35 and b 7 5 Evaluate -a -b Evaluate -a for a 35 -b Evaluate -a for a 35 and b 7 -b Evaluate -a for a 35 and b 7 -b Evaluate -a for a 35 and b 7 -b 35 Evaluate -a for a 35 and b 7 -b 35 7 Evaluate -a for a 35 and b 7 -b 35 5 7 Evaluate a b Evaluate a for a 15 b Evaluate a for a 15 and b 5 b Evaluate a for a 15 and b 5 b Evaluate a for a 15 and b 5 b 15 Evaluate a for a 15 and b 5 b 15 5 Evaluate a for a 15 and b 5 b 15 5 3 Evaluate -a b Evaluate -a for a 15 b Evaluate -a for a 15 and b 5 b Evaluate -a for a 15 and b 5 b Evaluate -a for a 15 and b 5 b 15 Evaluate -a for a 15 and b 5 b 15 5 Evaluate -a for a 15 and b 5 b 15 5 3 Evaluate a -b Evaluate a for a 15 -b Evaluate a for a 15 and b 5 -b Evaluate a for a 15 and b 5 -b Evaluate a for a 15 and b 5 -b 15 Evaluate a for a 15 and b 5 -b 15 5 Evaluate a for a 15 and b 5 -b 15 5 3 That is a a a b b b The set of rational numbers is the set of numbers a b Where a and b are integers and b is not equal to 0. The set of rational numbers contains the whole numbers, the integers, the positive and negative fractions. The set of rational numbers contains the whole numbers, the integers, the positive and negative fractions. The set of rational numbers contains the whole numbers, the integers, the positive and negative fractions. The set of rational numbers contains the whole numbers, the integers, the positive and negative fractions. We picture the rational numbers on a number line as follows: -5 0 5 We picture the rational numbers on a number line as follows: -5 -2.7 0 5 We picture the rational numbers on a number line as follows: -5 -2.7 0 5 We picture the rational numbers on a number line as follows: -5 -2.7 3 2 0 5 We picture the rational numbers on a number line as follows: -5 -2.7 3 2 0 5 We picture the rational numbers on a number line as follows: -5 -2.7 3 2 0 1.5 5 We picture the rational numbers on a number line as follows: -5 -2.7 3 2 0 1.5 5 We picture the rational numbers on a number line as follows: -5 -2.7 3 2 0 1.5 4 5 We picture the rational numbers on a number line as follows: -5 -2.7 3 2 0 1.5 4 5 5 Graph 2 -5 0 5 5 Graph 2 -5 0 5 2 5 5 Graph 2 -5 0 5 2 5 Graph 3.2 -5 0 5 Graph 3.2 -5 3.2 0 5 Graph 3.2 -5 3.2 0 5 13 Graph 8 -5 0 5 13 Graph 8 -5 0 13 8 5 13 Graph 8 -5 0 13 8 5 Convert to decimal notation 5 8 Divide 8 5.000 Divide . 8 5.000 Divide .6 8 5.000 Divide .6 8 5.000 48 Divide .6 8 5.000 48 2 Divide .6 8 5.000 48 20 Divide .62 8 5.000 48 20 Divide .62 8 5.000 48 20 16 Divide .62 8 5.000 48 20 16 4 Divide .62 8 5.000 48 20 16 40 Divide .62 5 8 5.000 48 20 16 40 Divide .62 5 8 5.000 48 20 16 40 40 Divide .62 5 8 5.000 48 20 16 40 40 0 Convert to decimal notation 5 8 .625 Convert to decimal notation 7 11 Divide 11 7.000 Divide . 11 7.000 Divide .6 11 7.000 Divide .6 11 7.000 66 Divide .6 11 7.000 66 4 Divide .6 11 7.000 66 40 Divide .6 3 11 7.000 66 40 Divide .6 3 11 7.000 66 40 33 Divide .6 3 11 7.000 66 40 33 7 Divide .6 3 11 7.000 66 40 33 70 Divide .6 3 6 11 7.000 66 40 33 70 Divide .6 3 6 11 7.000 66 40 33 70 66 Divide .6 3 6 11 7.000 66 40 33 70 66 4 Divide .6 3 6 36 11 7.000 66 40 33 70 66 4 Divide .6 3 6 36 11 7.000 66 40 33 70 66 4 Each rational number can be expressed in either terminating or repeating decimal notation. The following are other examples to show how each rational number can be named using fraction The following are other examples to show how each rational number can be named using fraction or decimal notation: The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 3 8 4 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 3 8 8.75 4 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 27 100 3 8 8.75 4 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 27 0.27 100 3 8 8.75 4 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 27 0.27 100 3 8 8.75 4 13 6 The following are other examples to show how each rational number can be named using fraction or decimal notation: 0 0 8 27 0.27 100 3 8 8.75 4 13 2.1 6 6 Each rational number has a point on the number line. Each rational number has a point on the number line. However, there are some points on the line for which there are no rational numbers. Each rational number has a point on the number line. However, there are some points on the line for which there are no rational numbers. These points correspond to what are called Each rational number has a point on the number line. However, there are some points on the line for which there are no rational numbers. These points correspond to what are called irrational numbers. What kinds of number are irrational? What kinds of number are irrational? One example is the number pi, What kinds of number are irrational? One example is the number pi, which is used in finding the area and the circumference of a circle: What kinds of number are irrational? One example is the number pi, which is used in finding the area and the circumference of a circle: A r 2 What kinds of number are irrational? One 2 example is the A r number pi, which is used in finding the area and the C 2 r circumference of a circle: Another example of an irrational number is the square root of 2, Another example of an irrational number is the square root of 2, named 2 It is the length of the diagonal of a square with sides of length l. It is the length of the diagonal of a square with sides of length l. It is the length of the diagonal of a square with sides of length l. 1 It is the length of the diagonal of a square with sides of length l. 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 c 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 c 2 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 c 1 2 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 c 1 1 2 2 2 2 1 1 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 c 1 1 2 2 2 1 1 2 c 2 2 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 c 1 1 2 2 2 1 1 2 c 2 c 2 2 It is the length of the diagonal of a square with sides of length l. c a b 2 2 c 1 1 2 c 1 1 2 2 2 2 1 2 c 2 c 2 2 1 It is also the number that when multiplied by itself gives 2—that is It is also the number that when multiplied by itself gives 2—that is 2 It is also the number that when multiplied by itself gives 2—that is 2 2 It is also the number that when multiplied by itself gives 2—that is 2 2 2 1.4 is an approximation of 2 1.4 is an approximation of because (1.4)2 = 1.96 2 1.4 is an approximation of because (1.4)2 = 1.96 2 1.41 is a better approximation of 2 1.4 is an approximation of because (1.4)2 = 1.96 2 1.41 is a better approximation of because (1.41)2 = 1.9881 2 1.4 is an approximation of because (1.4)2 = 1.96 2 1.41 is a better approximation of because (1.41)2 = 1.9881 1.414 is a better approximation of 2 2 1.4 is an approximation of because (1.4)2 = 1.96 2 1.41 is a better approximation of because (1.41)2 = 1.9881 1.414 is a better approximation of 2 because (1.414) = 1.99996164 2 2 Decimal notation for rational numbers either Decimal notation for rational numbers either terminates Decimal notation for rational numbers either terminates or repeats. Decimal notation for rational numbers either terminates or repeats. Decimal notation for irrational numbers neither terminates Decimal notation for rational numbers either terminates or repeats. Decimal notation for irrational numbers neither terminates nor repeats. Some other examples of irrational numbers are Some other examples of irrational numbers are 3, Some other examples of irrational numbers are 3, 8 , Some other examples of irrational numbers are 3, 8 , 11 , Some other examples of irrational numbers are 3, 8 , 11 , 0.121221222122221… The rational numbers and the irrational numbers together correspond to all the points on a number line and make up what is called the real-number system. The set of Real Numbers — The set of all numbers corresponding to points on the number line. Real numbers are named in order on the number line, -5 0 5 Real numbers are named in order on the number line, with larger numbers named farther to the right. -5 0 5 Real numbers are named in order on the number line, with larger numbers named farther to the right. -5 0 5 Real numbers are named in order on the number line, with larger numbers named farther to the right. -5 0 5 Real numbers are named in order on the number line, with larger numbers named farther to the right. LARGER -5 0 5 Real numbers are named in order on the number line, with larger numbers named farther to the right. SMALLER -5 LARGER 0 5 For any two numbers on the line, the one to the left is SMALLER -5 LARGER 0 5 For any two numbers on the line, the one to the left is less than the one to the right. SMALLER -5 LARGER 0 5 We use the symbol < to mean “is less than.” We use the symbol < to mean “is less than.” The sentence -8<6 We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” The symbol > We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” The symbol > means “is greater than.” We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” The symbol > means “is greater than.” then sentences -8<6 We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” The symbol > means “is greater than.” then sentences -8<6 and -3>-7 We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” The symbol > means “is greater than.” then sentences -8<6 and -3>-7 are inequalities. We use the symbol < to mean “is less than.” The sentence -8<6 means “-8 is less than 6” The symbol > means “is greater than.” then sentences -8<6 and -3>-7 are inequalities. 2 0 5 10 2 0 9 5 10 2 0 2 9 5 10 2 0 2 9 5 9 10 2 0 2 9 5 9 10 2 0 2 9 5 9 10 2 9 0 2 5 9 10 2 9 0 2 5 9 10 7 0 7 3 0 7 -7 3 0 7 -7 3 0 3 7 -7 3 0 3 7 -7 3 0 3 7 3 -7 0 3 7 3 -7 0 3 6 0 6 12 0 6 12 0 6 6 -12 12 0 6 6 -12 12 0 6 6 -12 12 0 6 6 12 -12 0 6 6 12 -12 0 6 18 0 18 5 0 18 -18 5 0 18 -18 5 -5 0 18 -18 5 -5 0 18 -18 5 -5 0 18 5 -18 -5 0 18 5 -18 -5 0 2.7 -5 0 5 2.7 -5 32 0 5 2.7 32 2.7 -5 0 5 2.7 32 2.7 -5 0 5 32 2.7 2.7 -5 3 2 0 5 32 2.7 2.7 -5 3 2 0 5 3 2.7 2 2.7 -5 3 2 0 5 3 2.7 2 2.7 -5 3 2 0 5 1 .5 -5 0 5 2.7 1 .5 -5 0 5 2.7 1 .5 1 .5 -5 0 5 2.7 1 .5 1 .5 2.7 -5 0 5 2.7 1 .5 1 .5 2.7 -5 0 5 2.7 1 .5 1 .5 2.7 -5 0 5 1.5 2.7 1 .5 2.7 -5 0 5 1.5 2.7 1 .5 2.7 -5 0 5 1.38 0 0.5 1 .0 1 .5 2 .0 1.38 0 0.5 1.83 1 .0 1 .5 2 .0 1.38 1.83 1.38 0 0.5 1 .0 1 .5 2 .0 1.38 1.83 1.38 0 0.5 1 .0 1 .5 1.83 2 .0 1.38 1.83 1.38 0 0.5 1 .0 1 .5 1.83 2 .0 1.38 1.83 1.38 0 0.5 1 .0 1 .5 1.83 2 .0 1.38 1.83 1.38 0 0.5 1 .0 1 .5 1.83 2 .0 1.38 1.83 1.38 0 0.5 1 .0 1 .5 1.83 2 .0 3.45 -5 0 5 3.45 -5 1.32 0 5 3.45 1.32 3.45 -5 0 5 3.45 1.32 3.45 -5 1.32 0 5 3.45 1.32 3.45 -5 1.32 0 5 3.45 1.32 3.45 -5 1.32 0 5 3.45 1.32 3.45 -5 1.32 0 5 3.45 1.32 3.45 -5 1.32 0 5 4 -5 0 5 4 -5 0 0 5 4 0 4 -5 0 5 4 4 -5 0 0 0 5 4 4 -5 0 0 0 5 4 4 -5 0 0 0 5 4 0 4 -5 0 0 5 4 0 4 -5 0 0 5 5.8 -3 0 5 5.8 -3 0 0 5 5.8 0 0 -3 0 5 5.8 0 5.8 0 -3 0 5 5.8 0 5.8 0 0 5 5.8 0 5.8 0 -3 0 5 5.8 0 5.8 0 -3 0 5 5.8 0 5.8 0 -3 0 5 5 8 -3 0 5 5 8 -3 0 7.11 5 5 8 7.11 5 8 -3 0 5 5 8 7.11 7.11 5 8 -3 0 5 5 8 7.11 7.11 5 8 -3 0 5 5 8 7.11 7.11 5 8 -3 0 5 5 8 7.11 7.11 5 8 -3 0 5 5 8 7.11 7.11 5 8 -3 0 5 1 2 0.9 0.7 0.5 0.3 0.1 0 1 2 1 3 0.9 0.7 0.5 0.3 0.1 0 1 2 1 3 .5 0.9 0.7 0.5 0.3 0.1 0 1 2 .5 1 3 .33 0.9 0.7 0.5 0.3 0.1 0 1 2 .5 1 3 .33 1 2 0.9 0.7 0.5 0.3 0.1 0 1 2 .5 1 2 1 3 .33 1 3 0.9 0.7 0.5 0.3 0.1 0 1 2 1 2 1 3 1 3 0.9 0.7 0.5 0.3 0.1 0 1 2 1 2 1 3 1 3 0.9 0.7 0.5 0.3 0.1 0 1 2 1 2 1 3 1 3 0.9 0.7 0.5 0.3 0.1 0 1 2 1 2 1 3 1 3 0.9 0.7 0.5 0.3 0.1 0 2 3 5 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 5 3.0 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 . 5 3.0 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 .6 5 3.0 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 2.6 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 2 3 4 2.6 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 2 2.6 3 4 2.75 2.5 2.0 1.5 1.0 0.5 0 2 2 3 5 11 4 2 3 5 2.6 3 4 2.75 2.5 2.0 1.5 1.0 0.5 0 2 11 4 2 3 5 11 4 2 3 5 2.6 3 4 2.75 2.5 2.0 1.5 1.0 0.5 0 2 11 4 2 3 5 11 4 3 5 2.5 2.0 1.5 1.0 0.5 0 2 11 4 2 3 5 11 4 3 5 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 2 11 4 3 5 2.5 2.0 1.5 1.0 0.5 0 2 3 5 11 4 2 11 4 3 5 2.5 2.0 1.5 1.0 0.5 0 Write another inequality with the same meaning. Write another inequality with the same meaning. 2 Write another inequality with the same meaning. 2 Write another inequality with the same meaning. 2 3 Write another inequality with the same meaning. 2 3 3 Write another inequality with the same meaning. 2 3 3 Write another inequality with the same meaning. 2 3 3 2 Write another inequality with the same meaning. Write another inequality with the same meaning. 3 Write another inequality with the same meaning. 3 Write another inequality with the same meaning. 3 8 Write another inequality with the same meaning. 3 8 8 Write another inequality with the same meaning. 3 8 8 Write another inequality with the same meaning. 3 8 8 3 Write another inequality with the same meaning. Write another inequality with the same meaning. a Write another inequality with the same meaning. a Write another inequality with the same meaning. a 5 Write another inequality with the same meaning. a 5 5 Write another inequality with the same meaning. a 5 5 Write another inequality with the same meaning. a 5 5 a If b is a positive real number then If b is a positive real number then b If b is a positive real number then b If b is a positive real number then b 0 If a is a negative real number then If a is a negative real number then a If a is a negative real number then a If a is a negative real number then a 0 -4 0 5 3 -4 0 5 3 -4 0 5 3 5.4 -4 0 5 3 5.4 -4 0 5 3 5.4 -4 0 5 3 5.4 -4 true 0 5 3 3 3 3 3 3 true -5 0 5 5 -5 0 5 5 -5 0 5 5 1 23 -5 0 5 5 1 23 -5 0 5 5 1 23 -5 0 5 5 1 23 -5 false 0 5 Find the absolute value. 7 Find the absolute value.. 7 7 Find the absolute value. 12 Find the absolute value. 12 12 Find the absolute value. 0 Find the absolute value. 0 0 Find the absolute value.. 3 2 Find the absolute value. 3 2 3 2 Find the absolute value. 2.73 Find the absolute value. 2.73 2.73 L 1.2 Numbers 1 - 68