Scientific Notation for Very Large and Small Quantities
Numbers too big and too small to write
Some measurements are too big or too small to write comfortably in ordinary digits. The distance light travels in a year runs to thirteen digits, and the width of an atom hides behind a string of leading zeros. Scientific notation is a compact, standard way to write such numbers, and it makes comparing and calculating with them far easier. Every number is written as a single digit before the decimal point, possibly followed by more digits, multiplied by a power of ten.
Standard form: the one-to-ten rule
The exact rule for standard form is that a number is written as a times ten to the power n, where a is at least one but less than ten, and n is a whole number. That value of a is called the mantissa, and the condition that it sits between one and ten is what makes the form standard and unambiguous. The power of ten simply records how far, and in which direction, the decimal point must move to recover the ordinary number. Getting a into that one-to-ten window is the heart of every conversion.
Large numbers and positive powers
Large numbers use positive powers of ten. To write three hundred thousand in standard form, place the decimal point after the first non-zero digit, giving three, then count how many places it must move right to rebuild the original: five places, so three hundred thousand is three times ten to the fifth. A messier example, six million seven hundred and twenty thousand, becomes six point seven two times ten to the sixth. The exponent counts decimal places, not zeros, which matters whenever the number has non-zero digits in the middle.
Small numbers and negative powers
Small numbers use negative powers of ten, which connects directly to the idea that a negative exponent means a reciprocal. The number nought point zero zero zero three is three times ten to the negative four, because the decimal point must move four places to the right to turn three into the original tiny value. Likewise nought point zero zero zero zero zero four five is four point five times ten to the negative six. A negative exponent is the signal that you are looking at a small quantity, less than one, not a negative quantity.
Time scales at both extremes
Time gives natural examples at both extremes, which is exactly the kind of context this skill is built for. A nanosecond, used to time computer operations, is one times ten to the negative nine seconds, while a microsecond is one times ten to the negative six seconds. At the other end, the age of the universe is about thirteen point eight billion years, written one point three eight times ten to the tenth years. Expressed in plain digits these intervals are awkward; in scientific notation their relative sizes are obvious at a glance.
Calculating with the index laws
Calculating is where the notation truly earns its place, because the index laws from earlier in the year do the heavy lifting. To multiply two numbers in scientific notation, multiply the mantissas and add the powers of ten. Three times ten to the fourth, multiplied by two times ten to the third, is six times ten to the seventh. To divide, divide the mantissas and subtract the powers: six times ten to the eighth divided by three times ten to the fifth is two times ten to the third. The same exponent laws you applied to bare powers apply here without change.
Checking the mantissa and comparing
One habit guards against the most common error: always check that the mantissa really sits between one and ten when you finish. A result like thirty-five times ten to the fourth is not yet in standard form, because thirty-five is too big; it should be rewritten as three point five times ten to the fifth. Equally, nought point six times ten to the seventh has a mantissa below one and should become six times ten to the sixth. Comparing two numbers is then easy: the one with the larger power of ten is larger, and if the powers match you compare the mantissas.