Problem 21 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. Mount Kilimanjaro is the highest mountain in Africa. It is 5890 meters high.

Step-by-Step Solution

Answer

Answer: The height of Mount Kilimanjaro in scientific notation is 5.89 × 10^3 meters.

Step 1: Identify the given number we want to convert

The given number is the height of Mount Kilimanjaro (5890 meters).

Step 2: Move the decimal point to create a coefficient between 1 and 10

In this case, we need to move the decimal point in 5890 meters to the left 3 places, resulting in the coefficient 5.89.

Step 3: Multiply the coefficient by a power of 10

Since we moved the decimal point 3 places to the left, we need to multiply the coefficient (5.89) by 10 raised to the power of 3 (10^3).

Step 4: Write the number in scientific notation

In scientific notation, the height of Mount Kilimanjaro (5890 meters) can be written as 5.89 × 10^3 meters.

Key Concepts

Converting to Scientific Notation

Converting numbers to scientific notation is a way to express very large or very small numbers compactly and with ease. The process involves formulating a number between 1 and 10, known as the coefficient, and multiplying it by a power of 10.

For instance, Mount Kilimanjaro's height is a relatively large number, 5890 meters. To convert it into scientific notation, the decimal point is moved three places to the left, which gives us the coefficient 5.89. Then, because the decimal point has moved three places, this change is accounted for by multiplying 5.89 by 10 raised to the power of 3, which effectively shifts the decimal point back to its original position when multiplied out. Hence, the scientific notation for 5890 meters is written as \(5.89 \times 10^3\) meters. This not only simplifies the number but also makes it easy to handle in mathematical operations.

Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. In scientific notation, the exponent is always 10, as 10 is the base of our number system.

The number of times the decimal point is moved in the number becomes the exponent when converting to scientific notation. For example, when converting 5890 to scientific notation, the decimal point moves 3 places; therefore, we use 10 raised to the power of 3, or 10^3. The exponent here is 3, which is a positive integer because the original number was greater than 10. This system helps maintain the scale of numbers without writing out numerous zeros, thus making calculations and comprehension much simpler.

Decimal Places

The term 'Decimal places' refers to the position of digits to the right of the decimal point in a number. It's imperative to understand the concept of decimal places when working with scientific notation since the coefficient must be between 1 and 10, and the rest of the significant figures are placed after the decimal point.

Moving decimal points to create a coefficient in this range is crucial and should be done with precision. For example, with the height of Mount Kilimanjaro as 5890 meters, the '890' after the '5' becomes part of the decimal places once we write it in scientific notation, resulting in 5.890, or more precisely 5.89 when rounded to two decimal places. This precision is critical in scientific and mathematical communications where exactness matters.