How would each of the following numbers be represented in \(\mathrm{E}\) notation? A) \(3.287 \times 10^{6}\) B) \(-978.65 \times 10^{12}\) C) \(7.65491 \times 10^{-3}\) D) \(-58710.23 \times 10^{-4}\)

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is widely used by scientists, engineers, and mathematicians to handle very large or very small numbers with ease.

In scientific notation, a number is written as the product of two factors: a decimal number and a power of ten. The decimal number, usually between 1 and 10, is called the 'coefficient', and the power of ten is written as an exponent. For example, the number 2,000 can be written in scientific notation as \(2 \times 10^3\), where the coefficient is 2, and the exponent on the 10 is 3, indicating the decimal has been moved three places to the right.

Exponent notation is a convenient way to express repeated multiplication of the same factor. When you have a number, say 'b', raised to a power 'n', written as \(b^n\), this tells us that 'b' is multiplied by itself 'n' times. Exponents can be positive, negative, or zero, each altering the base in different ways.

For positive exponents, like \(b^3\), it means \(b \times b \times b\). For negative exponents, such as \(b^{-3}\), it represents one over the base raised to the corresponding positive power, so \(b^{-3} \equiv \frac{1}{b^3}\). An exponent of zero means the value is always 1, no matter what 'b' is (except when 'b' is zero), as in \(b^0 = 1\).

Negative Exponents and E-Notation

Specifically relating to the textbook exercise, when dealing with E-notation, negative exponents are used to indicate very small numbers. Step 3 of the original exercise illustrated this by converting \(7.65491 \times 10^{-3}\) into an acceptable E-notation without further adjustment, because the coefficient was already between 1 and 10, and the exponent was a valid integer.

Floating-point representation is a system for approximating real numbers that cannot be exactly represented in a base system. It allows us to write a very large range of numbers in a standardized and compact form, making it particularly useful in computing and electronic devices.

In many programming languages and computational tools, you might see numbers represented in floating-point format, which is akin to scientific notation in the way it splits a number into parts. These parts consist of a sign (positive or negative), an exponent, and a significand (similar to the coefficient in scientific notation).

In the context of the textbook solutions, floating-point representation would allow for the standardized form of numbers with distinct precision levels. For example, in Step 4 of the problem, the number \(-58710.23 \times 10^{-4}\) is adjusted to \(-5.871023 \times 10^0\), where the exponent is zero and the significand is \(-5.871023\), easily indicating that the value is precise to six decimal places.