Problem 32 - Technical Mathematics with Calculus

Question

Convert each number from enginecring notation to decimal notation. $$6.37 \times 10^{3}$$

Step-by-Step Solution

Answer

The number in decimal notation is 6370.

Step 1: Understanding Engineering Notation

Engineering notation is a version of scientific notation in which the exponent of ten must be divisible by three. The given number is already in this form, and the task is to convert it to standard decimal notation.

Step 2: Converting to Decimal Notation

The given number is 6.37 multiplied by 10 to the power of 3. To convert it to decimal notation, shift the decimal point in the base number 3 places to the right, because the exponent of 10 is positive and equal to 3.

Step 3: Writing the Result

After moving the decimal point 3 places to the right, the number becomes 6370. The final step is to write down the result in standard decimal form which is 6370.

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It's widely used in science, engineering, and mathematics to make calculations easier and to provide a clear representation of numbers. The structure of a number in scientific notation is a combination of a coefficient and an exponent. The coefficient is a number greater than 1 and less than 10, and it is multiplied by 10 raised to an exponent.

For example, the number 2000 can be represented in scientific notation as \(2 \times 10^3\). Here '2' is the coefficient and '3' is the exponent, which means that the decimal point is moved three places to the right. This notation not only simplifies writing but also aids in understanding the scale and precision of numbers. Understanding scientific notation is crucial for converting between forms of numerical representation.

Decimal Conversion

Converting a number from scientific or engineering notation to decimal notation involves adjusting the position of the decimal point according to the exponent. When the exponent is positive, you move the decimal point to the right, indicating a larger number, and when it is negative, to the left, indicating a smaller number.

In our exercise, the given number in engineering notation is \(6.37 \times 10^3\). The exponent '3' tells us to move the decimal point three places to the right, resulting in 6370. The process is straightforward and requires careful attention to the direction and number of places the decimal is shifted. It is helpful to remember that the exponent corresponds to the number of zeroes following the digit in whole numbers when converting to decimal form.

Exponents

Exponents represent how many times a number, known as the base, is multiplied by itself. The exponent is written as a superscript number. For instance, \(10^3\) (read as 'ten to the third power' or 'ten cubed') means that 10 is multiplied by itself three times: \(10 \times 10 \times 10 = 1000\).

In the context of scientific and engineering notation, exponents are used to scale the base number up or down, making it easier to work with extremely large or small values. Positive exponents indicate a larger number, while negative exponents indicate a fraction. Mastery of exponents is essential for anyone working with scientific notation or performing mathematical operations involving powers of numbers.