Vocabulary Terms
1. Notation on the Calculator: 1.388e-2
The notation "1.388e-2" represents a number in scientific notation, more specifically, it can be read as \(1.388 \times 10^{-2}\). In this case, "e" stands for "exponent," so "e-2" indicates that we are multiplying by \(10^{-2}\).
2. Why does the calculator create this notation?
Calculators use this notation primarily for two reasons:
- Space Efficiency: It allows for compact representation of very large or very small numbers without using excessive digits, making it easier to read and manage calculations.
- Precision Maintenance: It accurately represents numbers that may otherwise be cumbersome to display fully or that require many digits, ensuring that the precision of the values is maintained.
3. Why come up with a new notation like Scientific Notation? What is the point?
Scientific notation simplifies calculations and communication of very large or very small numbers. The benefits include:
- Simplicity: It allows easier multiplication and division of large numbers by focusing on the exponents.
- Clarity: It reduces the risk of errors in reading or writing very large or small values.
- Standardization: It provides a common framework for scientists, mathematicians, and engineers, facilitating clearer communication of numeric data.
Problems
1. Positive Integer Exponent in Scientific Notation
A positive integer exponent tells you how many places to move the decimal point to the right when converting from scientific notation to standard form. For example, \(5.0 \times 10^3\) means you move the decimal 3 places to the right, resulting in 5000.
Example:
- \(3.2 \times 10^4 = 32,000\)
2. Negative Integer Exponent in Scientific Notation
A negative integer exponent tells you how many places to move the decimal point to the left when converting from scientific notation to standard form. For example, \(4.5 \times 10^{-3}\) means you move the decimal 3 places to the left, resulting in 0.0045.
Example:
- \(2.1 \times 10^{-2} = 0.021\)
3. Write 872,010,000,000 in Scientific Notation
To convert, you place the decimal after the first non-zero digit and count how many places you move it to the left.
- \(872,010,000,000 = 8.7201 \times 10^{11}\)
4. Write 0.00000000026 in Scientific Notation
To convert, you place the decimal after the first non-zero digit and count how many places you move it to the right.
- \(0.00000000026 = 2.6 \times 10^{-10}\)
5. A company reported a net income for the quarter as $2.75 billion. Last quarter, the income was reported as $1.76 billion. How much did the income increase?
To find the increase in income, subtract the last quarter's income from the current quarter's income.
Operation:
- Subtraction
Reasoning: To find the increase, I will calculate \(2.75 , \text{billion} - 1.76 , \text{billion}\).
Calculations:
- \(2.75 - 1.76 = 0.99\)
Conclusion: The income increased by $0.99 billion, or $990 million.