Scientific Notation

1Summary 重點摘要

Learn to write very large and very small numbers using scientific notation, no need to write many zeros.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What is scientific notationScientific notation (also called standard form) writes any number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Its value shows with extremely large or small numbers: the Earth–Sun distance of about 150000000 km becomes 1.5 × 10⁸ km, and a bacterium of diameter about 0.000002 m becomes 2 × 10⁻⁶ m. Without it, such numbers need long strings of zeros that are easy to miscount and hard to compare. There are only two rules: a must lie between 1 (included) and 10 (excluded), and n equals the number of places the decimal point moved — positive when moved left, negative when moved right.
2. Step 2: Writing large numbersWriting a large number in scientific notation takes three steps. For 300000: first, move the decimal point from the far right to just after the first non-zero digit, giving 3.00000; second, count the places moved — from 300000. to 3. is 5 places; third, write 3 × 10⁵. Practice again: the speed of light is about 300000000 m/s; the point moves 8 places to sit after the 3, giving 3 × 10⁸ m/s. Mind the choice of a: the remaining non-zero digits become the decimal part of a, so 2760000 = 2.76 × 10⁶, not 2 × 10⁶ — dropping the 76 would be a serious approximation error.
3. Step 3: Writing small numbersNumbers below 1 work the same way, except the exponent is negative. For 0.000045: first, move the point to just after the first non-zero digit (4), giving 4.5; second, count the places — from 0.000045 to 4.5 the point moved 5 places to the right; third, moving right means a negative exponent, so the answer is 4.5 × 10⁻⁵. One more: for 0.0032 the point moves 3 places right to give 3.2, so the answer is 3.2 × 10⁻³. Memory aid: large numbers take positive exponents, small numbers negative; the direction of movement is opposite to the sign — shrinking the number (point left) gives a positive exponent, growing it (point right) a negative one.
4. Step 4: Converting back (positive index)Converting back with a positive exponent means shifting the point right. For 2.5 × 10⁴: since 10⁴ = 10000, we get 2.5 × 10000 = 25000; equivalently, move the point in 2.5 four places right, padding with zeros. Two more: 3.06 × 10⁵ = 306000 (five places right), and 7 × 10² = 700 (a as a whole number is fine as long as 1 ≤ a < 10). This direction appears in astronomy and geography: the Earth's radius of about 6.4 × 10³ km is 6400 km. After converting, count the zeros against the original as a quick self-check against shifting one place too many or too few.
5. Step 5: Converting back (negative index)A negative exponent means shifting the point left. For 3.7 × 10⁻³: since 10⁻³ = 11000, we get 3.7 × 11000 = 0.0037 — the same as moving the point in 3.7 three places left. Two more: 4.5 × 10⁻⁵ = 0.000045 (five places left, four zeros in front), and 1.2 × 10⁻¹ = 0.12. Such numbers fill scientific measurement: a dust particle's mass may be about 7.5 × 10⁻¹⁰ kg, and a red blood cell's diameter about 8 × 10⁻⁶ m. When writing, follow the order '0., then the zeros, then the digits of a'; afterwards verify that the zeros between the point and the first non-zero digit are exactly one fewer than the exponent.
6. Step 6: Rules and checkingConsolidate two core rules. First, a always lies between 1 and 10 (including 1, excluding 10): 1234 correctly becomes 1.234 × 10³; 12.34 × 10² gives the same value but breaks the rule, and 0.5 × 10⁴ must be rewritten as 5 × 10³. Second, large numbers take positive n, small numbers negative n, and |n| equals the number of places the point moved. There are two self-checks: multiply the answer back to see the original return — 1.234 × 10³ = 1234; and develop a feel for magnitude — 1.234 × 10³ should sit at the 'thousands' scale, so if your conversion lands in the tens of thousands or the hundreds, the exponent is wrong and needs re-checking.
7. Step 7: Common mistakes summaryThree errors dominate scientific notation. First, an out-of-range a: writing 450 as 45 × 10¹ or 0.45 × 10³ instead of 4.5 × 10²; always confirm 1 ≤ a < 10 after writing. Second, a flipped exponent sign: writing 0.0007 as 7 × 10⁴ (it is 7 × 10⁻⁴) or 70000 as 7 × 10⁻⁴ (it is 7 × 10⁴); remember 'the bigger the number, the bigger the exponent'. Third, dropping the decimal part of a: writing 375000 as 3 × 10⁵ loses 75000 entirely; a must keep every non-zero digit — 375000 = 3.75 × 10⁵. Practising a handful of forward and backward conversions before an exam cuts these errors sharply.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Form: a × 10ⁿ
• a between 1 and 10
• Large n positive, small n negative
• Move decimal point and count places
• No need to write many zeros