Scientific Notation and Significant Figures

Scientific notation writes any number as a number from one up to, but not including, ten, multiplied by a power of ten. The distance to the Sun is one hundred and forty-nine million six hundred thousand kilometres. Slide the decimal point until one digit is in front of it: one point four nine six. The point moved eight places, so the power is ten to the eight. Fourteen point nine six times ten to the seven is the same size, but it is not scientific notation, because the front number is bigger than ten. Check by size: ten to the eight is a hundred million, and one and a half of those is a hundred and fifty million.

Numbers between zero and one use a negative power. For nought point nought nought nought nought five two, slide the point right to just after the first non-zero digit: five point two. It moved five places, and because the number is smaller than one, the power is negative five. Check it the other way: ten to the minus five is one hundred-thousandth, and five point two of those is the number we started with.

To compare two numbers in scientific notation, look at the powers of ten first. Three point two times ten to the five is three hundred and twenty thousand; nine point eight times ten to the four is ninety-eight thousand. The bigger power wins, even though nine point eight looks bigger than three point two. Only when the powers match do you compare the front numbers. With negative powers, remember that minus three is bigger than minus five, so ten to the minus three is the larger.

Significant figures start at the first non-zero digit. In nought point nought nought four nought eight one six, the first significant figure is the four. Three significant figures are four, nought, eight, and the next digit is a one, so it rounds to nought point nought nought four nought eight. Zeros sandwiched between non-zero digits count. For seven million two hundred and forty-nine thousand six hundred to two significant figures, keep seven and two, look at the four, round down, and fill with zeros so the number stays the same size: seven million two hundred thousand.

To multiply numbers in scientific notation, multiply the front numbers and add the powers. Three times five is fifteen, and ten to the four times ten to the six is ten to the ten. Fifteen times ten to the ten is the right size but not the right form, so write fifteen as one point five times ten, which adds one more to the power: one point five times ten to the eleven. Dividing works the same way: divide the front numbers and subtract the powers. Quiz and exercises now.