Problem 17 - Technical Mathematics with Calculus

Question

Convert each number from scientific notation to decimal notation. $$2.85 \times 10^{3}$$

Step-by-Step Solution

Answer

The number in decimal notation is 2850.

Step 1: Identify the Components of Scientific Notation

Recognize that the given number is in scientific notation, which has the form \(a \times 10^{n}\), where \(1 \leq |a| < 10\) and \(n\) is an integer. Here, \(a = 2.85\) and \(n = 3\).

Step 2: Determine the Shift of the Decimal Point

The exponent \(n = 3\) indicates that the decimal point must be shifted 3 places to the right.

Step 3: Convert to Decimal Notation

Moving the decimal point 3 places to the right gives \(2850\).

Key Concepts

Scientific Notation

Scientific notation is a systematic way to express large or small numbers. This method involves two main components: a coefficient and an exponent. The coefficient is a number usually between 1 and 10, and the exponent is an integer which tells us how many places to move the decimal point.

In scientific notation, numbers are written in the form of \(a \times 10^{n}\), where \(1 \leq |a| < 10\) and \(n\) is an integer. This form is particularly useful for simplifying calculations in science and engineering, as it allows for easy multiplication and division of extremely large or small quantities by adjusting the exponent.

Decimal Notation

Decimal notation is the standard form for representing numbers that most people are familiar with. In this system, the value of each digit is determined by its position relative to the decimal point. The digits to the left of the decimal point represent whole numbers, while those to the right represent fractions or parts of a whole.

Converting from scientific to decimal notation enables us to understand the size of a number in a more intuitive way, as it becomes explicit how many places the decimal point has been moved.

Exponents

Exponents are used to denote repeated multiplication of a base number. In the expression \(10^{n}\), the number 10 is the base and \(n\) is the exponent, signifying that 10 is to be multiplied by itself \(n\) number of times.

When \(n\) is positive, we move the decimal point to the right, indicating a larger number, while a negative \(n\) means moving the decimal point to the left, indicating a smaller number. Understanding the role of exponents is essential for mastering scientific notation and performing mathematical conversions.

Mathematical Conversion

Mathematical conversion between scientific notation and decimal notation involves the movement of the decimal point as dictated by the exponent. The process can be broken into simple steps to ensure accuracy.

First, identify the coefficient and the exponent. Then, based on whether the exponent is positive or negative, shift the decimal point of the coefficient accordingly. Moving the decimal to the right indicates an increase in value, while shifting it to the left indicates a decrease. This methodical approach allows for the precise conversion between these two numerical expressions.