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Chapter 1: Problem 26

Change the number from ordinary notation to engineering notation. $$925,000,000,000$$

Short Answer

Expert verified
925,000,000,000 is 925 \times 10^9 in engineering notation.

Step by step solution

01

Understand Engineering Notation

Engineering notation is a variation of scientific notation, where the exponent of 10 must be a multiple of 3. This means numbers are expressed as a power of 10 with the exponent in increments of 3.
02

Identify the Decimal Point Placement

Start with the number 925,000,000,000. In ordinary notation, the decimal point is at the end of the number. To convert to engineering notation, you need to move this decimal point by increments of 3.
03

Move the Decimal Point

Move the decimal point 9 places to the left to convert the number to fit within an exponent of 10 that is a multiple of 3. This results in moving the decimal such that: \( 925,000,000,000 = 925 \times 10^9 \).
04

Confirm Correct Exponent

Ensure that the power of 10 is a multiple of 3, which in this case, it is (since 9 is a multiple of 3). The number 925 is appropriate because it is less than 1000, which is a requirement for engineering notation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scientific Notation
Scientific notation is a mathematical method of writing large or small numbers in a more concise form. It is especially useful in science and engineering, where very large or very small numbers are common. The format for scientific notation is:
  • It's composed of a base number and an exponent of 10.
  • The base number is typically between 1 and less than 10.
  • The exponent indicates how many times the base number should be multiplied or divided by 10.
For instance, the number 5,000,000 can be rewritten as:\[5 \times 10^6\]Here, 5 is the base number, and the exponent 6 indicates that 5 should be multiplied by 10 six times to reach 5,000,000. This compact form makes it easier to work with extremely large or small numbers without writing all the zeros.
Exponentiation
Exponentiation is a mathematical operation that involves raising a number (called the base) to the power of an exponent. It indicates how many times the base is used as a factor in a product. For example:
  • In the expression \(2^4\), 2 is the base, and 4 is the exponent.
  • So, \(2^4\) means \(2 \times 2 \times 2 \times 2\), which equals 16.
In scientific and engineering notation, exponentiation is crucial because it helps express numbers in a simplified form by using powers of 10. This reduces the need to write extensive sequences of zeros, making calculations and comparisons more manageable.
Decimal Point Movement
Decimal point movement involves shifting the decimal point in a number to change its value while adjusting the corresponding power of ten. This concept is key in both scientific and engineering notation.
  • To convert a number into scientific or engineering notation, you move the decimal point to create a new base number.
  • In engineering notation specifically, the decimal must be moved in increments of three places to ensure the resulting exponent is a multiple of 3.
For example, consider the number 925,000,000,000. Starting with the decimal at the end, you move it 9 places to the left:\[925,000,000,000 = 925 \times 10^9\]Here, every 3-place shift represents an increment in the exponent by 3. Such decimal movement is crucial for accurately representing numbers in engineering notation, ensuring a clean and manageable expression.

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Most popular questions from this chapter

Assume that all numbers are approximate unless stated otherwise. The percent of alcohol in a certain car engine coolant is found by performing the calculation \(\frac{100(40.63+52.96)}{105.30+52.96} .\) Find this percent of alcohol. The number 100 is exact.

Perform the indicated multiplications. Simplify the expression \(\left(T^{2}-100\right)(T-10)(T+10),\) which arises when analyzing the energy radiation from an object.

Perform the indicated operations and attach the correct units to your answers. $$\left(28.2 \frac{\mathrm{ft}}{\mathrm{s}}\right)(9.81 \mathrm{s})$$

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. Describe the values of \(x\) and \(y\) for which (a) \(|x+y|=|x|+|y|\) and (b) \(|x-y|=|x|+|y|\).

Perform the indicated operations. The first number is approximate, and the second number is exact. $$8.62 \div 1728$$
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