Question

When expressed in standard scientific notation, numbers greater than 1 will have (positive/negative) exponents, whereas numbers less than 1 will have (positive/negative) exponents.

Short answer

In standard scientific notation, numbers greater than 1 will have positive exponents, whereas numbers less than 1 will have negative exponents.

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers in the form of \[a\times 10^b\], where \(1 \le |a| < 10\) and \(b\) is an integer.

Analyzing the relationship between the number and the exponent

When a number is greater than 1, in order to express it in scientific notation, we need to move the decimal point to the left. This effectively divides the number by a power of 10 and the exponent is positive. For example, the number 300 can be written in scientific notation as \(3 \times 10^2\). In contrast, when a number is less than 1, we need to move the decimal point to the right. This means that we are multiplying the number by a power of 10, and the exponent becomes negative. For example, the number 0.03 can be written in scientific notation as \(3 \times 10^{-2}\).

Concluding the relationship between the size of the number and the exponent

In standard scientific notation: - For numbers greater than 1, the exponent will be positive. - For numbers less than 1, the exponent will be negative.

Key concepts

Standard Scientific Notation

Standard scientific notation is a concise method for expressing very large or very small numbers. It simplifies analysis and computation, especially in fields such as science and engineering. The format for standard scientific notation is \(a \times 10^b\), where \(a\) is a number greater than or equal to 1 but less than 10, and \(b\) is an integer. It's essential that \(a\) is not 10 or more, as this would necessitate moving the decimal point, changing the power of 10.

In essence, this system creates a shorthand to avoid writing out a long string of zeros. To correctly convert a number to scientific notation, you would:

• Identify the most significant digit in the number.
• Place the decimal point immediately after this digit.
• Count the number of places you moved the decimal point; this becomes the exponent \(b\).
• If the original number is greater than 1, the exponent is positive. If it’s less than 1, the exponent is negative.

For example, the number 53,000 would be expressed in scientific notation as \(5.3 \times 10^4\) because the decimal is moved 4 places to the left to place it after the 5 (the most significant digit).

Positive and Negative Exponents

Positive and negative exponents play a crucial role in determining the size of the number in scientific notation. To understand this:

• When the exponent is positive as in \(10^3\), it signifies the number of times 10 is multiplied by itself, which in this case is \(10 \times 10 \times 10 = 1,000\).
• A negative exponent like \(-3\) in \(10^{-3}\) means that we divide 1 by 10 multiplied by itself 3 times, effectively \(\frac{1}{10 \times 10 \times 10} = 0.001\).

So, a positive exponent indicates a large number, and a negative exponent indicates a small number relative to 1. It's an inverse relationship: positive to increase, negative to decrease. This fundamental understanding is crucial as it aids in comprehending the magnitude and scale of numbers, particularly in scientific and mathematical contexts.

Expressing Large and Small Numbers

Expressing large and small numbers efficiently is crucial in various scientific disciplines. Without scientific notation, it would be cumbersome to write out all the digits of very large numbers like the speed of light (299,792,458 meters per second) or ultra-small figures such as the mass of an electron (0.000000000000000000000000000000910938356 kilograms).

In scientific notation:

• The speed of light becomes \(2.99792458 \times 10^8\) meters per second.
• The mass of an electron is expressed as \(9.10938356 \times 10^{-31}\) kilograms.

By using scientific notation, you can see the relative magnitude of numbers at a glance, which aids in understanding and calculations. For instance, when multiplying two numbers in scientific notation, you can simply add their exponents, simplifying the process significantly. This notation allows scientists, engineers, and students alike to work with extreme quantities without error or confusion.