Scientific Notation

Numbers in scientific notation have two parts: a mantissa and an exponent which are multiplied together.

The mantissa has a value that is greater than or equal to 1 and less than 10 (ie it has a single non-zero digit before a decimal point).

The exponent is a power of 10 by which the mantissa is multiplied so that it has the same value as the original number.

When multiplying (or dividing) by a power of ten the digits of a number shift in the place value column table; however, on paper this is not easy to do (or visualise) so the (relative) effect on the decimal point is considered instead:

• Multiplying a number by a power of 10 shifts the digits left which has the effect of shifting the decimal point to the right;
• Dividing a number by a power of 10 shifts the digits right which has the effect of shifting the decimal point to the left.

If a number is followed by a unit (metres, litres, grams, etc) then the number is converted and the unit follows the converted number.

To convert a number to scientific notation:

1. Put the decimal point after the first non-zero digit;
2. Count how many digits the decimal point has to move to get back to its original position;
3. If the decimal point has to move left make this count negative;
4. Remove trailing zeros;
5. Put the number with the decimal point after the first non-zero digit and without the trailing zeros multiplied by 10 to the power of the count (made negative if necessary).

Also, if the number is not quite in scientific notation, that is, it has a number which does not have the decimal point after the first-non-zero digit but is multiplied by a power of 10, follow the above on the number before the x 10^... and then multiply the ten to the powers together by adding the powers

Examples:

123 → 1.23 × 10^2

123000 → 1.23 × 10^5

12.3 → 1.23 × 10^1

1.23 → 1.23 × 10^0

0.123 → 1.23 × 10^-1

0.000123 → 1.23 × 10^-4

12.3 × 10^5 = (1.23 × 10^1) × 10^5 = 1.23 × (10^1 × 10^5) = 1.23 × 10^(1 + 5) = 1.23 × 10^6

123 × 10^-3 = (1.23 × 10^2) × 10^-3 = 1.23 × (10^2 × 10^-3) = 1.23 × 10^(2 + -3) = 1.23 × 10^-1

To convert a number from scientific notation to normal notation move the decimal point the number of digits that is the power of the ten, inserting zeros if the digits run out. If the power is positive move the decimal point to the right; if negative to the left.

Examples

1.23 × 10^1 = 12.3

1.23 × 10^5 = 12300

1.23 × 10^0 = 1.23 (the decimal point moves 0 or no digits)

1.23 × 10^-1 = .123 = 0.123*

1.23 × 10^-5 = 0.0000123

* a zero is normally written before a leading decimal point to highlight the decimal point being there as leading zeros are not normally written.

Metric units include a multiplier in the letter preceding the unit, for example in km (kilometres) the k prefix means kilo- which is "× 1000" or "× 10^3".

To convert SI units to scientific notation, they are effectively in "not-quite" scientific notation and so the above is used by fist converting the unit multiplier to a power of 10. Examples:

1.23 km = 1.23 × 10^3 m

12.3 km = (1.23 × 10^1) × 10^3 m = 1.23 × (10^1 × 10^3) m = 1.23 × 10^4 m

123 mg = (1.23 × 10^2) × 10^-3 g = 1.23 × (10^2 × 10^-3) g = 1.23 × 10^-1 g

The most common Si prefix multipliers are:

pico- (p) = × 10^-12

nano- (n) = × 10^-9

micro- (µ) = × 10^-6

milli- (m) = × 10^-3

centi- (c) = × 10^-2

deci- (d) = × 10^-1

deka- (da) = × 10^1

hecto- (h) = × 10^2

kilo- (k) = × 10^3

mega- (M) = × 10^6

giga- (G) = × 10^9

Tera- (T) = × 10^12

Seventeen and two hundred twenty millionths (17.00022*) in standard form = 1.700022 × 101

*this is the standard notation