Standard form
Introduced| English | Meaning |
|---|---|
| standard form/ˈstændəd fɔːm/ | A positive number written a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. |
| integer/ˈɪntɪdʒə/ | A whole number that may be positive, negative or zero. |
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.
Standard form route
Follow a large or small number into a x 10^n form.
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.
In standard form, the front number A must satisfy 1 ≤ A < 10.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
By definition, A is between 1 (inclusive) and 10 (exclusive) — exactly one digit before the decimal point.
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).
Quick method. Write the first non-zero digit, then a decimal point, then the remaining significant digits. Count the places you moved: that's $n$.
A galaxy: huge distances are written compactly in standard form
Write 4 500 000 as A × 10ⁿ. What is the power n?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
4 500 000 = 4.5 × 10⁶, so n = 6.
Write 0.00072 in standard form A × 10ⁿ. What is n?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
0.00072 = 7.2 × 10⁻⁴, so n = −4.
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.
- $(8 \times 10^7) \div (4 \times 10^3) = 2 \times 10^4$.
Adjust if needed. After multiplying, the front number might be $\geq 10$. Adjust: $15 \times 10^3 = 1.5 \times 10^4$.
(3 × 10⁵) × (2 × 10⁻²) = A × 10³. What is A?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
3 × 2 = 6 and 10⁵ × 10⁻² = 10³, so the answer is 6 × 10³.
(8 × 10⁷) ÷ (4 × 10³) = A × 10ⁿ. What is n?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
8 ÷ 4 = 2 and 10⁷ ÷ 10³ = 10⁴, so n = 4.
Which is 5 × 10³ × 3 × 10² in correct standard form?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
5 × 3 = 15 and 10³ × 10² = 10⁵, giving 15 × 10⁵. Adjust: 1.5 × 10⁶.
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.
Calculate (3.2 × 10⁵ + 7 × 10⁴) / 10⁵.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Use the same power: 3.2 + 0.7 = 3.9.
You've got it
- standard form $= A \times 10^n$ with $1 \leq A < 10$
- big numbers → positive $n$; small numbers $(< 1)$ → negative $n$
- multiply: fronts $\times$, powers $+$; divide: fronts $\div$, powers $-$
- $4\,500\,000 = 4.5 \times 10^6$; $\;0.000\,72 = 7.2 \times 10^{-4}$
Write from a power of 5
- Write 890 000 in standard form.
- State the power of 10.
Write 890 000 as A times 10 to a power. What is that power?
The decimal moves five places.
What is the power?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
890 000 is 8.9 times 10 to the fifth.