Standard form · ⁨科学记数法⁩

From atoms to galaxies

• The width of a hydrogen atom is about $0.000\,000\,000\,106$ metres.
• The distance to the nearest star is about $40\,000\,000\,000\,000$ metres.
• Writing numbers like that is painful — and error-prone. Standard form 科学记数法 fixes both problems.

The format $A \times 10^n$

• Standard form writes any number as $A \times 10^n$, where:
• $1 \leq A < 10$ (the "front number" has exactly one digit before the decimal point),
• $n$ is an integer 整数 (positive for big numbers, negative for small ones).

Each tick is ten times bigger than the last. Standard form tells you which tick a number sits near.

Converting big and small numbers

• Big numbers: count how many places the decimal moves left → positive power.
• $4\,500\,000 = 4.5 \times 10^6$ (point moves 6 places left).
• Small numbers (less than 1): count how many places the decimal moves right → negative power.
• $0.000\,72 = 7.2 \times 10^{-4}$ (point moves 4 places right).

A galaxy: huge distances are written compactly in standard form

Calculating in standard form

• Multiply: multiply the fronts, add the powers.
• $(3 \times 10^5) \times (2 \times 10^{-2}) = 6 \times 10^3$.
• Divide: divide the fronts, subtract the powers.
• $(8 \times 10^7) \div (4 \times 10^3) = 2 \times 10^4$.

Why scientists love standard form

• The speed of light: $3.0 \times 10^8$ m/s — three hundred million metres every second.
• A DNA helix width: $2.0 \times 10^{-9}$ m — two billionths of a metre.
• Compare like quantities with the same units. A length of $2.0\times10^{-6}$ m is $1000$ times a length of $2.0\times10^{-9}$ m; speed cannot be compared with width.

Adding needs a common power

• For addition, first write both numbers with the same power of ten: $3.2\times10^5+7\times10^4=(3.2+0.7)10^5=3.9\times10^5$.
• Renormalise after calculating: $(8\times10^3)(4\times10^2)=32\times10^5=3.2\times10^6$. Core calculations in standard form are on calculator Paper 3.

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