Gre math Review 1 Arithmetic
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1.5 Real NumbersThe set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line. Arithmetic Figure 1 below is a number line. Arithmetic Figure 1 Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. As shown in Arithmetic Figure 1 the negative numbers negative 0.4, negative 1, negative 3 over 2, negative 2, the negative square root of 5, and negative 3 are to the left of 0 and the positive numbers 1 over 2, 1, the positive square root of 2, 2, 2.6, and 3 are to the right of 0. Only the number 0 is neither negative nor positive. A real number x is less than a real number y if x is to the left of y on the number line, which is written as x followed by the less than symbol followed by y. A real number y is greater than x if y is to the right of x on the number line, which is written as y followed by the greater than symbol followed by x. For example, the number line in Arithmetic Figure 1 shows the following three less than or greater than relationships. Relationship 1: the negative square root of 5 is less than negative 2 Relationship 2: the fraction 1 over 2 is greater than 0 Relationship 3: 1 is less than the positive square root of 2, which is less than 2 To say that a real number x is between 2 and 3 on the number line means that x is greater than 2 and x is less than 3, which can also be written as the double inequality 2 is less than x, which is less than 3. The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality 2 is less than x, which is less than 3 is often used to represent that interval. Note that the endpoints of the interval, 2 and 3, are not included in the interval. Sometimes one or both of the endpoints are to be included in an interval. The following inequalities represent four types of intervals, depending on whether the endpoints are included. Interval type 1: 2 is less than x, which is less than 3 Interval type 2: 2 is less than or equal to x, which is less than 3 Interval type 3: 2 is less than x, which is less than or equal to 3 Interval type 4: 2 is less than or equal to x, which is less than or equal to 3 There are also four types of intervals with only one endpoint, each of which consists of all real numbers to the right or to the left of the endpoint, perhaps including the endpoint. The following inequalities represent these types of intervals. Interval type 1: x is less than 4
The entire real number line is also considered to be an interval. The distance between a number x and 0 on the number line is called the absolute value of x, written as the absolute value symbol around the number x. Therefore, the absolute value of 3 = 3 and the absolute value of negative 3 = 3 because each of the numbers 3 and negative 3 is a distance of 3 from 0. Note that
Example A: the absolute value of the positive square root of 5 = the positive square root of 5 Example B: the absolute value of negative 23 = negative, open parenthesis, negative 23, close parenthesis, which is equal to 23 Example C: the absolute value of negative 10.2 = 10.2 Here are twelve general properties of real numbers that are used frequently. In each property a, b, and c are real numbers. Property 1: a + b = b + a and a b = b a
Property 2: open parenthesis, a + b, close parenthesis, + c = a +, open parenthesis, b + c and open parenthesis, a b, close parenthesis, times c = a times, open parenthesis, bc, close parenthesis. For example, open parenthesis 7 + 3, close parenthesis, + 8 = 7 +, open parenthesis, 3 + 8, close parenthesis, which is equal to 18 and open parenthesis, 7 times the positive square root of 2, close parenthesis, times the positive square root of 2 = 7 times, open parenthesis, the positive square root of 2 times the positive square root of 2, close parenthesis, which is equal to 7 times 2, or 14. Property 3: a times, open parenthesis, b + c, close parenthesis, = a b + bc For example, 5 times, open parenthesis, 3 + 16, close parenthesis, = 5 times 3, +, 5 times 16, which is equal to 95. Property 4: a + 0 = a, a times 0 = 0, and a times 1 = a. Property 5: If a b = 0, then either a = 0 or b = 0 or both. For example, if negative 2b = 0, then b = 0. Property 6: Division by 0 is not defined; for example, 5 divided by 0, negative 7 over 0, and 0 over 0 are undefined. Property 7: If both a and b are positive, then both a + b and a b are positive. Property 8: If both a and b are negative, then a + b is negative and a b is positive. Property 9: If a is positive and b is negative, then a b is negative. Property 10:
Property 11: the absolute value of a times the absolute value of b = the absolute value of the quantity a times b. For example, the absolute value of 5 times the absolute value of negative 2 = the absolute value of the quantity 5 times negative 2, which is equal to the absolute value of negative 10, which is equal to 10. Property 12: If a is greater than 1, then a squared is greater than a. If 0 is less than b, which is less than 1, then b squared is less than b. For example, 5 squared = 25, which is greater than 5, but, open parenthesis, 1 over 5, close parenthesis, squared = 1 over 25, which is less than 1 over 5. Yüklə 0,76 Mb. Dostları ilə paylaş:
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