Standard Form
Standard Form
Standard form (also called standard index form or scientific notation) is a compact way of writing very large or very small numbers. Without it, values like the distance from Earth to the nearest star (about 40 200 000 000 000 000 m) or the diameter of a virus (about 0.000 000 1 m) are unwieldy and error-prone to write. Standard form removes that problem by expressing every number as a power of 10 multiplied by a coefficient.
The format is \( A \times 10^n \) where \( A \) is the coefficient and \( n \) is the index. Two strict conditions must hold: \( 1 \leq A < 10 \) (exactly one non-zero digit before the decimal point) and \( n \) must be an integer (positive, negative or zero).
Converting an Ordinary Number to Standard Form
Move the decimal point until exactly one non-zero digit remains to the left of it — this gives the coefficient \( A \). Count the number of places moved: that count is \( n \). If the original number was large, the decimal moved left so \( n \) is positive. If the original number was small, the decimal moved right so \( n \) is negative.
Example: convert 0.000 072 to standard form.
Move the decimal point 5 places right to get 7.2 — so \( A = 7.2 \) and \( n = -5 \):
\( 0.000,072 = 7.2 \times 10^{-5} \)
Converting from Standard Form to an Ordinary Number
Reverse the process: if \( n \) is positive, move the decimal \( n \) places right (making the number larger). If \( n \) is negative, move it \( |n| \) places left (making the number smaller). Fill any gaps with zeros.
Example: \( 3.04 \times 10^6 \) → move decimal 6 places right → 3 040 000.
Calculating with Standard Form
For multiplication and division, handle the coefficient and the power of 10 separately, then recombine — adjusting if \( A \) falls outside \( [1, 10) \). For addition and subtraction, first convert both numbers to the same power of 10.
Worked Examples
Write 56 800 000 in standard form. Write \( 2.07 \times 10^5 \) as an ordinary number.
To standard form: 56 800 000 — decimal point is after the last zero. Move it left until one non-zero digit remains to the left: 5.68 (moved 7 places). So \( n = 7 \).
\[ 56,800,000 = 5.68 \times 10^7 \]From standard form: \( 2.07 \times 10^5 \) — positive index, move decimal 5 places right.
\[ 2.07 \times 10^5 = 207,000 \]Write 0.000 049 in standard form. Write \( 8.1 \times 10^{-3} \) as an ordinary number.
To standard form: Move decimal right until 4.9 — moved 5 places right, so \( n = -5 \).
\[ 0.000,049 = 4.9 \times 10^{-5} \]From standard form: \( 8.1 \times 10^{-3} \) — negative index, move decimal 3 places left.
\[ 8.1 \times 10^{-3} = 0.0081 \]Calculate \( (3 \times 10^4) \times (4 \times 10^6) \) and \( (2.4 \times 10^6) + (5 \times 10^5) \).
Multiplication: Handle coefficients and indices separately.
\[ (3 \times 10^4) \times (4 \times 10^6) = (3 \times 4) \times 10^{4+6} = 12 \times 10^{10} \]Since \( 12 geq 10 \), adjust: \( 12 \times 10^{10} = 1.2 \times 10^{11} \).
Addition: Convert to the same power of 10 first.
\[ 5 \times 10^5 = 0.5 \times 10^6 \] \[ (2.4 \times 10^6) + (0.5 \times 10^6) = 2.9 \times 10^6 \]Key Takeaways
- Standard form is \( A \times 10^n \) where \( 1 \leq A < 10 \) and \( n \) is an integer — both conditions are mandatory.
- Positive \( n \): large number (decimal moves right when converting). Negative \( n \): small number (decimal moves left).
- For multiplication: multiply coefficients and add indices. For division: divide coefficients and subtract indices.
- After multiplication or division, check whether \( A \) is still in \( [1, 10) \) — if not, adjust \( A \) and \( n \) accordingly.
- For addition and subtraction: express both numbers with the same power of 10 before combining the coefficients.