This paper introduces scientific notation as a concise method for expressing very large and very small numbers using powers of ten. It explains the three components of a number in scientific notation — the coefficient, the base, and the exponent — and provides a step-by-step guide for converting standard numbers into scientific notation. The paper also outlines the arithmetic rules governing multiplication, division, addition, and subtraction in scientific notation, and reviews the fundamental laws of exponents that underpin these calculations. Real-world examples, such as computer storage capacity, are used to illustrate the practical relevance of the notation.
Consider a practical example from everyday life: a computer hard disk holds 4 gigabytes of information — that is 4,000,000,000 bytes. Written in scientific notation, that is 4 × 109 bytes.
Scientific notation is used to write very large and very small numbers. While ordinary numbers are useful for everyday measurements, for large measurements such as astronomical distances, scientific notation offers a concise way of expressing those values. Because many large and small numbers consist of just a few significant digits plus many zeros, the power of ten can be used to shorten the written form of the number considerably.
A number written in correct scientific notation is made up of three parts. The first part, the coefficient, is a number between 1 and 10. The second part, the base, is always a power of ten. The third part is the exponent, which indicates how many decimal places the decimal point must be moved to convert the number back to standard notation. A negative exponent means that the decimal point is moved to the left during that conversion.
The mathematical formula for writing a number in scientific notation is:
n × 10x
where n is a number greater than or equal to 1 but less than 10, and x is an integer exponent of 10. You can read more about the role of exponentiation in mathematics to better understand how powers of ten function in this context.
Follow these two steps to convert large or small numbers into scientific notation:
Step 1. Place the decimal point after the first non-zero digit and drop any zeros that are not significant. (All nonzero digits are considered significant; so too are any digits that express the precision of a measurement rather than its magnitude.) The resulting value is the coefficient.
Step 2. Determine the exponent by counting the number of places the decimal point has moved from its new position to the end of the original number. Numbers smaller than 1 will have a negative exponent. For a helpful reference on significant figures and how they relate to measurement precision, consult a standard mathematics or chemistry resource.
Using scientific notation not only provides a concise way to write very large and very small numbers — it also simplifies calculations.
Multiplication: To multiply numbers written in scientific notation, multiply the coefficients together and add the exponents. The base remains 10.
Division: To divide numbers written in scientific notation, divide the coefficients and subtract the exponents. The base remains 10.
"Rules for multiplication, division, addition, and subtraction"
"Fundamental exponent laws used in calculations"
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