The real Number system
- 1. The Real Number System Real Numbers Rational Numbers ... -1.5, -1, -0.5, 0, 0.5, 1, 1.5, … Integers ... -3, -2, -1, 0, 1, 2, 3, … Whole Numbers 0, 1, 2, 3, 4, 5, 6, … Natural Numbers 1, 2, 3, 4, 5, 6, … ... 2, 1.816, 0, 1, 2, ...π− − Irrational Numbers 2, 3, , 11...π
- 2. The Real Number Line | | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5 Integers -5 -4 -3 -2 -1 0 1 2 3 4 5 Natural Numbers -5 -4 -3 -2 -1 0 1 2 3 4 5 Whole Numbers | | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5 -4.5 -2.5 0.33 1.7 2.9 4.1 Rational Numbers 2 5 π Irrational Numbers Remember that Irrational Numbers do not have an exact point on the number line.
- 3. Square Roots A number is a Perfect Square if it is the product of two Identical Factors. • =6 6 36 • =10 10 100 • =2 2 4 − • =−5 5 25 − • =−7 7 49 These are all perfect squares because they are the product of two identical factors.
- 4. Common Perfect Squares Here is a list of the first 20 perfect squares. I should probably try to remember those.
- 5. Evaluating Square Roots Evaluate the following radicals 16 25 9 81 400 225 169 529 4 5 3 9 A number is a Perfect Square if it is the product of two Identical Factors. 20 15 13 23
- 6. More Evaluating Radicals 121− 5.29± 16 81 2.89 12.25 − −11 ±2.3 4 9 − 1.7 3.5 These are all Rational Numbers 15 20 90 80 3.87 4.47 9.49 8.94 These are all Irrational Numbers
- 7. More Irrational Roots Irrational Roots will always fall between two consecutive integers. Between what two consecutive integers is each square root? 35 130− 217 Between 5 and 6 Between -12 and -11 Between 14 and 15 Asi De Facil That was easy between 144 2& 1 1− − between 25 & 36 between 196 & 225
- 8. Homework Page 20: 10 – 22 & 28 - 36 Even Numbers Only
- 9. Expressing a Decimal as a Rational Number Write the fraction as the given number over the place value. Reduce the resulting fraction if possible. 0.3 0.225 0.2375 0.0325 3 10 225 1,000 9 40 2,375 10,000 19 80 325 10,000 13 400 A rational number is any number that can be written in the form a b
- 10. Expressing Rational Numbers as a Decimal A rational number is any number that can be written in the form Common Fractions Decimal Fractions Written with a numerator and a denominator. 1 2 3 4 1 16 To express as a decimal, perform the division. .5= .75= .0625= The numerator is written after the decimal point and the denominator is indicated by the place value of the last digit. 0.5 0.75 0.0625 15 10 2 = = 75 10 3 0 4 = = 625 10,0 100 1 6 = = These are all terminating decimals. a b
- 11. Repeating Decimals The same number, or pattern of numbers, repeats. 1 3 0.3333333333= 0.3= 2 3 0.6666666667= 0.6= 4 9 0.4444444444= 0.4= 2 11 0.1818181818= 0.18= 4 33 0.1212121212= 0.12= 8 33 0.2424242424= 0.24= That was easy
- 12. Symbols of Inequality Symbol Example Written in Words > 9 2> 9 is greater than 2 < 2 9< 2 is less than 9 ≥ 9 2≥ 9 is greater than or equal to 2 ≤ 2 9≤ 2 is less than or equal to 9 ≠ 2 9≠ 2 is not equal to 9
- 13. Comparing Real Numbers Compare the following numbers with an inequality symbol. 17, 1 4 3 Write each as a decimal 4.1231, 4.3 Compare the decimals 4.1231 4.3< Write the answer with the original numbers 1 17 4 3 < 2 5 3 , 29 5.6 5., 3852 5.6 5.3852> 2 5 29 3 > 4 3 , 2 1.3 1., 4142 1.3 1.4142< 4 2 3 < 7 11 , 0.63 ,0.63 0.63 0.63 0.63> 7 0.63 11 >
- 14. Ordering Numbers from Least to Greatest Order the numbers in each group from least to greatest. 3.2 3.002 3.02 3.002 3.02 3.2 2.021 2.21 2.0021− − − 2.21 2.021 2.0021− − − 5 3 2 5 5 5 12 4 3 5.416 5.75 5.6 1 7 2 5 2.4 2 4 − − 0.5 2 2.2361 1.75 2.4− − 2 1.75 0.5 2.2361 2.4− − 5 2 3 5 5 5 12 3 4 7 1 2 5 2.4 4 2 − − 5.416 5.6 5.75
- 15. Homework Page 20: 38 - 50 Even Numbers Only