Question

Convert each number to engineering notation. $$3.58 \times 10^{2}$$

Short answer

\( 3.58 \times 10^{2} \) in engineering notation is \( 358 \times 10^{0} \).

Step-by-step solution

Understand Engineering Notation

Engineering notation is a version of scientific notation in which the exponent on 10 is divisible by 3, so units of measure can easily be switched among SI prefixes. An engineering notation number will look like this: 'A x 10^n' where n is an integer divisible by 3.

Adjust the Exponent to a Multiple of 3

To express the number in engineering notation, we need to shift the decimal point such that the exponent of 10 (currently 2) becomes a multiple of 3. We can decrease the exponent by 2 to get it to 0, which is divisible by 3, by moving the decimal point to the right by 2 places.

Rewrite the Number in Engineering Notation

By moving the decimal point 2 places to the right, we have the number 358. Since the exponent is now a multiple of 3 (0 is considered as a multiple of 3), the expression in engineering notation is: \( 358 \times 10^0 \).

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a shorthand form that makes them easier to read, compare, and perform calculations with. It consists of two parts: a coefficient and a power of ten. The coefficient must be a number greater or equal to 1 and less than 10, and it is followed by the letter 'x' (meaning 'times') and then a power of 10 written as '10' raised to an exponent.

For example, the number 6,700,000 can be written in scientific notation as \( 6.7 \times 10^{6} \), and the small number 0.000012 can be written as \( 1.2 \times 10^{-5} \). Notice how the exponent denotes the number of places the decimal point has been moved to the right (for a positive exponent) or to the left (for a negative exponent). This notation is especially useful in fields such as engineering and physics as it simplifies the arithmetic of very large or very small numbers.

SI Prefixes

The International System of Units (SI) uses a set of prefixes to denote multiples and submultiples of units. These prefixes are attached to unit names to indicate multiplication by a particular power of ten. For example, the prefix 'kilo-' indicates multiplication by \( 10^{3} \) (or one thousand), so one kilometer (km) is equal to one thousand meters. Other common SI prefixes include 'centi-' for \( 10^{-2} \) (one hundredth), 'milli-' for \( 10^{-3} \) (one thousandth), and 'mega-' for \( 10^{6} \) (one million).

Using SI prefixes, engineers and scientists can effortlessly convey the scale of measurements in a universally understood way. A microampere (\( \mu A \)), for instance, is one-millionth of an ampere, indicating that this unit is highly practical for expressing small quantities of electric current.

Exponents

Exponents are a shorthand notation for expressing repeated multiplication of the same number. For instance, \( 10^4 \) (read as 'ten to the fourth power') represents \( 10 \times 10 \times 10 \times 10 \). The number 10 is the 'base' and the number 4 is the 'exponent' or 'power'. Exponents can also be negative, which indicates division. For example, \( 10^{-2} \) equals \( 1 / (10 \times 10) \), which is 0.01.

In more complex scenarios, exponents can be fractions, zero, or even irrational numbers. The zero exponent rule states that any non-zero base raised to the power of zero is 1. This property is reflected in the engineering notation example where \( 3.58 \times 10^2 \) converted to \( 358 \times 10^0 \), with the exponent 0 confirming the coefficient is exactly 358.