Problem 99 - College Algebra with Modeling and Visualization

Question

Writing about Mathematics Suppose that a positive number \(a\) is written in scientific notation as \(a=b \times 10^{n}\), where \(n\) is an integer and \(1 \leq b \leq 10 .\) Explain what \(n\) indicates about the size of \(a\)

Step-by-Step Solution

Answer

The exponent \(n\) indicates the number's order of magnitude, showing how large or small it is.

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It expresses a number as the product of a coefficient and a power of ten. In the expression \(a = b \times 10^{n}\), \(b\) is a number between 1 and 10, and \(n\) is an integer.

Step 2: Identify the Role of 'n'

The integer \(n\) in the expression \(a = b \times 10^{n}\) determines the size of the number \(a\) by indicating how many times the base 10 is multiplied by itself. If \(n\) is positive, it shows the number of zeroes follows after the coefficient \(b\), making \(a\) a large number. If \(n\) is negative, it indicates how many times \(b\) is divided by 10, making \(a\) a small number.

Step 3: Example of Large and Small n

Suppose \(n = 4\). This means \(a = b \times 10^4\), which translates \(a\) into a larger number by moving the decimal point four places to the right. For example, if \(b = 3\), then \(a = 30000\). Conversely, if \(n = -3\), then \(a = b \times 10^{-3}\) translates into a smaller number by moving the decimal point three places to the left, like if \(b = 3\), then \(a = 0.003\).

Key Concepts

Integer Exponents

When we talk about integer exponents in scientific notation, we refer to the exponent "\(n\)" in the expression \(a = b \times 10^{n}\). This number indicates how many times 10 is multiplied by itself. Integer exponents can be both positive and negative, and they have a significant impact on the size of the number in scientific notation. For example:

• \(10^2\) means 10 multiplied by itself twice: 10 \(\times\) 10, which equals 100.
• \(10^0\) is equal to 1 because any number to the power of zero is one.

Understanding integer exponents is crucial as they help in determining whether a given number in scientific notation is large or small.

Decimal Form

Decimal form is a way of writing numbers without exponents, which is what we often use in day-to-day life. In scientific notation, converting to decimal form involves adjusting the decimal point of the coefficient "\(b\)" according to the integer exponent \(n\). Depending on the sign of \(n\), the decimal point is either moved to the right or left:

• If \(n\) is positive, move the decimal point to the right by \(n\) places. For example, \(b = 2.5\) and \(n = 3\) results in 2500 in decimal form.
• If \(n\) is negative, move the decimal point to the left by \(n\) places. So, \(b = 7.1\) and \(n = -2\) results in 0.071.

Understanding this conversion is essential to visualize scientific notation numbers in a familiar format.

Positive and Negative Exponents

In scientific notation, exponents can take on positive or negative values, and each conveys different information about the size of the number.
**Positive Exponents:**

• When \(n\) is positive, \(a = b \times 10^{n}\) represents a large number. Here, the decimal point is moved to the right, increasing the value.
• For example, \(3 \times 10^4\) results in 30,000. This occurs because there are four places after the decimal point where zeroes are added.

**Negative Exponents:**

• A negative \(n\) implies \(a = b \times 10^{-n}\), which represents a small number. Consequently, the decimal point moves to the left, reducing the value.
• For instance, \(5 \times 10^{-3}\) becomes 0.005 after moving the decimal three places to the left.

Grasping these concepts of positive and negative exponents is vital for mastering how scientific notation scales numbers up or down.