3.1 Scientific Notation & Significant Digits

3.1 Scientific Notation & Significant Digits

Competency 1 skill 4 is worded carefully: represent and perform operations with real number approximations with scientific notation, giving attention to significant digits. That final clause is the part most candidates skip, and it is explicitly in the blueprint.

Normalized form

A number is in scientific notation when it is written as

a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer.

The coefficient a must have exactly one nonzero digit before the decimal point.

The exponent sign tells you the size, not the sign of the number. A negative exponent means the value is between −1 and 1; it does not mean the number is negative. So −5.2 × 10⁴ is a large negative number, while 5.2 × 10⁻⁴ is a small positive number. Distractors trade on this constantly.

Also watch for non-normalized answers. 34.7 × 10⁵ is numerically equal to 3.47 × 10⁶, but only the second is in scientific notation. Items regularly offer the unnormalized version as a trap after a multiplication.

Multiplying and dividing

Multiply: multiply the coefficients, add the exponents.

(3.2 × 10⁵)(4.5 × 10⁻²) = (3.2 · 4.5) × 10⁵⁺⁽⁻²⁾ = 14.4 × 10³

The coefficient 14.4 is out of range, so renormalize: 14.4 = 1.44 × 10¹, giving 1.44 × 10⁴.

Divide: divide the coefficients, subtract the exponents.

(8.4 × 10⁻³) ÷ (2.1 × 10⁶) = (8.4 ÷ 2.1) × 10⁻³⁻⁶ = 4 × 10⁻⁹

If the coefficient quotient falls below 1 — say 2.4 ÷ 8 = 0.3 — renormalize upward: 0.3 × 10⁻⁵ = 3 × 10⁻⁶. Moving the decimal right decreases the exponent.

Adding and subtracting

You cannot add coefficients until the exponents match. Rewrite the smaller-exponent number to match the larger.

6.2 × 10⁵ + 3.4 × 10⁴ Rewrite: 3.4 × 10⁴ = 0.34 × 10⁵ Add: (6.2 + 0.34) × 10⁵ = 6.54 × 10⁵

Adding coefficients and exponents separately — producing 9.6 × 10⁹ — is the classic error and always appears as an option.

Counting significant digits

Significant digits express how precisely a measurement is known.

+---------------------------------------------------------------------------+
|                       Significant Digit Rules                             |
+---------------------------------------------------------------------------+
|  1. All nonzero digits are significant.            4,872 -> 4 sig digits  |
|  2. Zeros BETWEEN nonzero digits are significant.  40.07 -> 4 sig digits  |
|  3. LEADING zeros are NEVER significant.          0.0035 -> 2 sig digits  |
|  4. Trailing zeros AFTER a decimal point ARE.      2.500 -> 4 sig digits  |
|  5. Trailing zeros with NO decimal point are       4,500 -> ambiguous;    |
|     ambiguous; scientific notation resolves it.    4.5 x 10^3 -> 2        |
|                                                    4.500 x 10^3 -> 4      |
+---------------------------------------------------------------------------+

Rule 5 is the reason scientific notation exists for measurement work. Writing 4,500 leaves the reader unsure whether the measurement is precise to the hundreds or to the ones; writing 4.50 × 10³ states unambiguously that three digits are known.

Rounding rules differ by operation

This distinction is a genuine blueprint-level detail.

Multiplication and division: match the fewest significant digits.

(4.72 × 10³)(2.1 × 10²) = 9.912 × 10⁵. The factor 2.1 has only 2 significant digits, so the answer is reported as 9.9 × 10⁵.

Addition and subtraction: match the fewest decimal places.

14.7 + 3.42 + 0.006 = 18.126. The least precise addend, 14.7, is known only to the tenths place, so the sum is reported as 18.1.

Note that the second rule is about place value, not digit count: 0.006 has just one significant digit but is known to the thousandths, so it is not the limiting term.

Contextual items

Real-number-approximation items usually arrive dressed as science.

A cell measures 2.5 × 10⁻⁵ meters across. A slide holds a line of 8.0 × 10³ such cells end to end. How long is the line?

Multiply: (2.5)(8.0) = 20, and 10⁻⁵ · 10³ = 10⁻². So 20 × 10⁻² = 2.0 × 10⁻¹ meters, or 0.20 meters. Both factors carry 2 significant digits, so the answer carries 2.

Light travels 3.0 × 10⁸ m/s. How far does it travel in 4.0 × 10⁻⁶ seconds?

(3.0)(4.0) = 12 and 10⁸ · 10⁻⁶ = 10², so 12 × 10² = 1.2 × 10³ meters. A candidate who subtracts exponents gets 10¹⁴ — an answer wrong by twelve orders of magnitude, and always offered.