To transform numbers from decimal form to scientific notation, it must be remembered that the laws of exponents form the basis for calculations using powers.

Using the results of the previous chapter, the following whole numbers and decimals can be expressed as powers of 10:

A number N is in scientific notation when it is expressed as the product of a decimal number between 1 and 10 and some integer power of 10.

The steps for converting to scientific notation are as follows:

Step 1:Place the decimal immediately to the right of the left-most non-zero number.

Step 2:Count the number of digits between the old and new decimal point.

Step 3:If the decimal is shifted to the left, the exponent is positive. If the decimal is shifted to the right, the exponent is negative.

Let us examine the logic of this. Consider as an example the number 3750. The number will not be changed if it is multiplied by 1000 and divided by 1000 (the net effect is to multiply it by one). Then,

There is a division by 10 for each space the decimal point is moved to the left, which is compensated for by multiplying by 10. Similarly, for a number such as .0037, we multiply the number by 10 for each space the decimal point is moved to the right. Thus, the number must be divided by 10 for each space.

Example 1:

Circulating water flows at 440,000 gallons per minute. Express this number in scientific notation.

Solution:

Example 2:

Express 0.0000247 in scientific notation.

Solution:

Example 3:

Express 34.2 in scientific notation.

Solution: