Perform operations in scientific notation, choose units of appropriate size and interpret technology output

Learning goal: Perform operations with numbers expressed in scientific notation, including problems where both decimal notation and scientific notation are used; use scientific notation and choose units of appropriate size for measurements of very large or very small quantities; and interpret scientific notation that has been generated by technology.

What scientific notation is. A positive number is in scientific notation when it is written as a × 10ⁿ where a is a decimal with exactly one nonzero digit before the decimal point, so 1 ≤ a < 10, and n is an integer. A negative number carries its sign in front, as in −6.84 × 10⁶, so the rule for the size of a applies to the number without its sign, and zero is simply written 0. The number 6,840,000 is 6.84 × 10⁶, and 0.000032 is 3.2 × 10⁻⁵. Unlike the single-digit estimates of the previous lesson, scientific notation keeps as many significant digits as the measurement has; 6.84 × 10⁶ is exact for 6,840,000. Converting between decimal notation and scientific notation is a matter of moving the decimal point and counting: moving it 6 places left multiplies by 10⁶ in compensation; moving it 5 places right multiplies by 10⁻⁵. A form such as 68.4 × 10⁵ has the same value but is not in scientific notation because 68.4 is not between 1 and 10; it is renormalized by moving the point one place left and raising the exponent by one.

Multiplying and dividing. Multiply the decimal parts and add the exponents, using the product rule for powers of 10: (3 × 10⁴)(2 × 10⁶) = 6 × 10¹⁰. If the product of the decimal parts leaves the 1-to-10 range, renormalize: (4 × 10⁵)(5 × 10³) = 20 × 10⁸ = 2 × 10⁹. Divide the decimal parts and subtract the exponents: (8 × 10⁷)/(2 × 10³) = 4 × 10⁴, and (2 × 10⁵)/(8 × 10²) = 0.25 × 10³ = 2.5 × 10². The renormalizing step is where most errors occur, because 0.25 × 10³ and 2.5 × 10² are equal in value but only the second is in scientific notation.

Adding and subtracting. Two numbers in scientific notation can be added by adding their decimal parts only when their powers of 10 are the same, in the same way that place values must line up in column addition; 10² + 10³ is a perfectly good sum, 1100, but it cannot be found by adding 1 + 1. To add 3.2 × 10⁵ and 4.5 × 10⁴, rewrite one so both have the same exponent: 4.5 × 10⁴ = 0.45 × 10⁵. Then 3.2 × 10⁵ + 0.45 × 10⁵ = 3.65 × 10⁵. To subtract 2.6 × 10⁻⁴ from 7.1 × 10⁻³, rewrite 2.6 × 10⁻⁴ as 0.26 × 10⁻³, then 7.1 − 0.26 = 6.84, giving 6.84 × 10⁻³. Adding the exponents here would be wrong; exponents add for multiplication only. Check any sum by converting to decimals: 320,000 + 45,000 = 365,000 = 3.65 × 10⁵.

Mixing decimal notation and scientific notation — a fully worked model. A sample contains 0.00045 grams of a substance, and a second sample contains 3.2 × 10⁻⁴ grams. Find the total and how many times the first is of the second. First convert the decimal to scientific notation: 0.00045 = 4.5 × 10⁻⁴. Total: 4.5 × 10⁻⁴ + 3.2 × 10⁻⁴ = 7.7 × 10⁻⁴ grams, which is 0.00077 g. Ratio: (4.5 × 10⁻⁴)/(3.2 × 10⁻⁴) = 4.5/3.2 = 1.40625, about 1.4 times. When exponents are equal the ratio is just the ratio of the decimal parts. Now a division with mixed forms: 2,500,000 ÷ (5 × 10²). Convert 2,500,000 to 2.5 × 10⁶, then (2.5 × 10⁶)/(5 × 10²) = 0.5 × 10⁴ = 5 × 10³ = 5000. Check: 2,500,000 ÷ 500 = 5000. The rule is to convert everything to one notation before operating; working in decimal notation is fine for small exponents and scientific notation is safer for large ones.

Choosing units of appropriate size. Scientific notation and unit choice work together. Suppose a mid-ocean ridge spreads at 2.5 centimeters per year. In meters that is 2.5 × 10⁻² m per year; in kilometers it is 2.5 × 10⁻⁵ km per year; in millimeters it is 25 mm per year. For a yearly rate, millimeters per year gives a whole number that is easy to picture and to measure. Over 1 × 10⁶ years the ridge would spread 25 mm × 10⁶ = 2.5 × 10⁷ mm, and here millimeters are a poor unit for the answer; convert to kilometers: 2.5 × 10⁷ mm = 2.5 × 10⁴ m = 25 km. A good guide is to choose the unit that gives a short number that is easy to picture, often one roughly between 1 and 1000; the exponent in scientific notation tells you how far the current unit is from that range, and each factor of 10³ is one step up the metric ladder (mm to m to km). The guide has limits: kilometers are the largest everyday metric unit of length, so very large distances are still given in kilometers even when the number is in the hundreds of thousands. A very small measurement such as 8 × 10⁻⁶ m is better reported as 8 micrometers, and a very large one such as 4 × 10⁸ m as 4 × 10⁵ km. The spreading rate above is a stated value for the exercise, not a measurement of any particular ridge.

Interpreting scientific notation generated by technology. Many calculators, spreadsheets and measuring instruments display scientific notation in a compact form that needs no superscripts, called E notation; some devices can show a raised exponent instead, and the setting varies. A display of 3.2E-5 means 3.2 × 10⁻⁵ = 0.000032; 4.5E12 means 4.5 × 10¹² = 4,500,000,000,000; and a spreadsheet cell showing 1.5E+09 means 1.5 × 10⁹ = 1,500,000,000. The letter E stands for "times ten to the exponent", and the number after it is the exponent, sign included. Two misreadings are common. First, 3.2E-5 does not mean 3.2 to the power of −5, and it does not mean 3.2 × e; it means 3.2 × 10⁻⁵. Second, many devices switch to this display on their own when a result is too large or too small for the ordinary display (when this happens depends on the device and its settings), so a result of 6.84E-3 from a calculation such as 7.1 × 10⁻³ − 2.6 × 10⁻⁴ is the same number as 0.00684 and should be translated back to units and a sensible number of digits before it is reported. The table in this lesson shows four displays and their meanings.

Digits a display cannot add. A display may round or cut off the digits a device has computed, and even the digits it shows say nothing about the inputs: a result cannot be more precise than the measurements that went into it. If 3.2 × 10⁵ m and 4.5 × 10⁴ m were each measured only to two digits, their sum 3.65 × 10⁵ m carries a third digit that the measurements never supported, and reporting about 3.7 × 10⁵ m is the honest choice. The display 3.65E5 shows what the arithmetic produced, not how precise the inputs were; deciding how many digits to report belongs to the person, not the device.

Misconceptions. First, exponents add in multiplication but never in addition; 10³ + 10³ is 2 × 10³, not 10⁶. Second, 0.5 × 10⁴ is a correct value but not scientific notation; rewrite as 5 × 10³. Third, a negative exponent does not make the quantity negative. Fourth, a number in scientific notation has not been rounded unless you round it; 6.84 × 10⁶ is exactly 6,840,000. Fifth, in the E display the sign after E belongs to the exponent, so 2.0E-3 is a small positive number, 0.002.

Worked multiplication and division: Compute (6 × 10⁻³)(4 × 10⁸) and (6 × 10⁻³)/(4 × 10⁸). Product: 6 × 4 = 24 and −3 + 8 = 5, so 24 × 10⁵ = 2.4 × 10⁶. Quotient: 6/4 = 1.5 and −3 − 8 = −11, so 1.5 × 10⁻¹¹. Check the product in decimals: 0.006 × 400,000,000 = 2,400,000 = 2.4 × 10⁶.

Worked sum with different exponents: Add 9.6 × 10⁷ and 8.5 × 10⁶. Rewrite 8.5 × 10⁶ as 0.85 × 10⁷. Then 9.6 + 0.85 = 10.45, giving 10.45 × 10⁷, which must be renormalized to 1.045 × 10⁸. Check: 96,000,000 + 8,500,000 = 104,500,000 = 1.045 × 10⁸.

Worked mixed notation with units: A tank holds 3.6 × 10⁴ liters and 8,500 liters are drained. The remaining volume is 3.6 × 10⁴ − 0.85 × 10⁴ = 2.75 × 10⁴ liters, or 27,500 L. Because 8,500 is short in decimal form, the same calculation works as 36,000 − 8,500 = 27,500, and the two routes agree. Choosing to keep liters is appropriate because 27,500 is readable; converting to kiloliters gives 27.5 kL, which is also appropriate and shorter.

Worked interpretation of a display: A calculator shows 7.2E-4 after dividing a mass of 0.36 grams by 500. Read it as 7.2 × 10⁻⁴ g = 0.00072 g. Check: 0.36 ÷ 500 = 0.00072. For a mass this small, milligrams are a better unit: 0.00072 g × 1000 = 0.72 mg. Reporting "0.72 mg" communicates the same value with a readable number.

Worked unit choice for a rate: A stated spreading rate of 2.5 cm per year is 2.5 × 10⁻² m/yr, 2.5 × 10⁻⁵ km/yr and 25 mm/yr. For describing yearly motion, 25 mm/yr is the appropriate unit because the number is small and whole. For the distance after 4 × 10⁶ years, 25 × 4 × 10⁶ = 1 × 10⁸ mm = 1 × 10⁵ m = 100 km, and kilometers are the appropriate unit for the result.

Operations record: Make a four-row table for the operations multiply, divide, add and subtract. In each row write the rule for the decimal parts, the rule for the exponents, one example with two numbers in scientific notation, and one example where one number is given in decimal notation and must be converted first. Mark every step where renormalizing was needed to keep the decimal part between 1 and 10. Below the table, translate four technology displays of your own choosing, two with negative exponents, into scientific notation and into decimal notation.

Independent performance: Write a short measurement scenario of your own involving one very large and one very small quantity, for example a distance and a thickness, with values you state as assumptions. Perform at least one operation of each kind on those values, showing decimal-part and exponent work separately, and for each answer choose a unit of appropriate size and justify the choice in a sentence. Include one calculation whose result you present first as a calculator-style display such as 4.8E-7 and then in ordinary notation with a unit. Success criteria: every answer is in correct scientific notation with the decimal part between 1 and 10, every addition or subtraction shows the exponents being matched first, every unit choice is justified by the size of the resulting number, and every display is translated correctly with the sign of the exponent respected.