Question

1. express the following sets in braces. ( )

a) the set of natural numbers less than 10.
b) the set of odd integers from -5 to 5.
c) the set of all whole numbers.
d) the set of all integers.
e) a set of three rational numbers from 1 to 2

e) a set of four real numbers from -2 to 1

1. list all possible subsets for the set {2, 4, 6}

1. the following sequences of numbers each have a limit. that is, they gradually get closer and closer to a specific number, called the limit. identify the limit of each of the following sequences

a) 7.1, 7.01, 7.001, 7.0001, …
b) 0.3, 0.33, 0.333, 0.3333, …
c) -2.6, -2.66, -2.666, -2.6666, …

1. express each of the following in standard notation.

a) $8 \times 10^5$
b) $-2.9 \times 10^8$
c) $6.54 \times 10^{-3}$
d) $9 \times 10^{-11}$

1. express each of the following values in scientific notation.

a) 5000000000
b) 0.000000007
c) -65200000
d) 0.000000000008745

Explanation:

Let's solve these problems one by one. We'll start with problem 8 (Express each of the following in Standard Notation) and problem 9 (Express each of the following values in scientific notation).

Problem 8: Express in Standard Notation

Scientific notation is in the form \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer. To convert to standard notation, we move the decimal point according to the exponent \( n \):

Part (a): \( 8 \times 10^5 \)

• Step 1: Recall that \( 10^5 = 100000 \).
• Step 2: Multiply 8 by 100000: \( 8 \times 100000 = 800000 \).

Part (b): \( -2.9 \times 10^8 \)

• Step 1: \( 10^8 = 100000000 \).
• Step 2: Multiply -2.9 by 100000000: \( -2.9 \times 100000000 = -290000000 \).

Part (c): \( 6.54 \times 10^3 \)

• Step 1: \( 10^3 = 1000 \).
• Step 2: Multiply 6.54 by 1000: \( 6.54 \times 1000 = 6540 \).

Part (d): \( 9 \times 10^{-11} \)

• Step 1: \( 10^{-11} = 0.00000000001 \) (11 zeros after the decimal before the 1).
• Step 2: Multiply 9 by \( 10^{-11} \): \( 9 \times 0.00000000001 = 0.00000000009 \).

Problem 9: Express in Scientific Notation

Scientific notation is \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer. We need to move the decimal point to get a number between 1 and 10, then count the number of places moved (positive for large numbers, negative for small numbers).

Part (a): \( 5000000000 \)

• Step 1: Move the decimal point 9 places to the left to get \( 5.0 \).
• Step 2: So, \( 5000000000 = 5 \times 10^9 \).

Part (b): \( 0.000000007 \)

• Step 1: Move the decimal point 9 places to the right to get \( 7.0 \).
• Step 2: Since we moved right, the exponent is -9. So, \( 0.000000007 = 7 \times 10^{-9} \).

Part (c): \( -65200000 \)

• Step 1: Move the decimal point 7 places to the left to get \( -6.52 \).
• Step 2: So, \( -65200000 = -6.52 \times 10^7 \).

Part (d): \( 0.00000000008745 \)

• Step 1: Move the decimal point 11 places to the right to get \( 8.745 \).
• Step 2: Since we moved right, the exponent is -11. So, \( 0.00000000008745 = 8.745 \times 10^{-11} \).

Final Answers

Problem 8 (Standard Notation):

a) \( 8 \times 10^5 = \boldsymbol{800000} \)
b) \( -2.9 \times 10^8 = \boldsymbol{-290000000} \)
c) \( 6.54 \times 10^3 = \boldsymbol{6540} \)
d) \( 9 \times 10^{-11} = \boldsymbol{0.00000000009} \)

Problem 9 (Scientific Notation):

a) \( 5000000000 = \boldsymbol{5 \times 10^9} \)
b) \( 0.000000007 = \boldsymbol{7 \times 10^{-9}} \)
c) \( -65200000 = \boldsymbol{-6.52 \times 10^7} \)
d) \( 0.00000000008745 = \boldsymbol{8.745 \times 10^{-11}} \)

Answer:

Let's solve these problems one by one. We'll start with problem 8 (Express each of the following in Standard Notation) and problem 9 (Express each of the following values in scientific notation).

Problem 8: Express in Standard Notation

Scientific notation is in the form \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer. To convert to standard notation, we move the decimal point according to the exponent \( n \):

Part (a): \( 8 \times 10^5 \)

• Step 1: Recall that \( 10^5 = 100000 \).
• Step 2: Multiply 8 by 100000: \( 8 \times 100000 = 800000 \).

Part (b): \( -2.9 \times 10^8 \)

• Step 1: \( 10^8 = 100000000 \).
• Step 2: Multiply -2.9 by 100000000: \( -2.9 \times 100000000 = -290000000 \).

Part (c): \( 6.54 \times 10^3 \)

• Step 1: \( 10^3 = 1000 \).
• Step 2: Multiply 6.54 by 1000: \( 6.54 \times 1000 = 6540 \).

Part (d): \( 9 \times 10^{-11} \)

• Step 1: \( 10^{-11} = 0.00000000001 \) (11 zeros after the decimal before the 1).
• Step 2: Multiply 9 by \( 10^{-11} \): \( 9 \times 0.00000000001 = 0.00000000009 \).

Problem 9: Express in Scientific Notation

Scientific notation is \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer. We need to move the decimal point to get a number between 1 and 10, then count the number of places moved (positive for large numbers, negative for small numbers).

Part (a): \( 5000000000 \)

• Step 1: Move the decimal point 9 places to the left to get \( 5.0 \).
• Step 2: So, \( 5000000000 = 5 \times 10^9 \).

Part (b): \( 0.000000007 \)

• Step 1: Move the decimal point 9 places to the right to get \( 7.0 \).
• Step 2: Since we moved right, the exponent is -9. So, \( 0.000000007 = 7 \times 10^{-9} \).

Part (c): \( -65200000 \)

• Step 1: Move the decimal point 7 places to the left to get \( -6.52 \).
• Step 2: So, \( -65200000 = -6.52 \times 10^7 \).

Part (d): \( 0.00000000008745 \)

• Step 1: Move the decimal point 11 places to the right to get \( 8.745 \).
• Step 2: Since we moved right, the exponent is -11. So, \( 0.00000000008745 = 8.745 \times 10^{-11} \).

Final Answers

Problem 8 (Standard Notation):

a) \( 8 \times 10^5 = \boldsymbol{800000} \)
b) \( -2.9 \times 10^8 = \boldsymbol{-290000000} \)
c) \( 6.54 \times 10^3 = \boldsymbol{6540} \)
d) \( 9 \times 10^{-11} = \boldsymbol{0.00000000009} \)

Problem 9 (Scientific Notation):

a) \( 5000000000 = \boldsymbol{5 \times 10^9} \)
b) \( 0.000000007 = \boldsymbol{7 \times 10^{-9}} \)
c) \( -65200000 = \boldsymbol{-6.52 \times 10^7} \)
d) \( 0.00000000008745 = \boldsymbol{8.745 \times 10^{-11}} \)