Scientific Notation and Place Value

Scientific notation makes very large and very small numbers easier to read and calculate. The coefficient shows the significant digits; the exponent shows how far the decimal point's place value shifts.

Convert a large number

Write 4,820,000 as 4.82 × 10⁶. The decimal moved six places left to make a coefficient between 1 and 10, so the exponent is positive six. Multiplying by one million restores the original number.

Convert a small decimal

Write 0.00073 as 7.3 × 10⁻⁴. The decimal moves four places right to create 7.3, so the exponent is negative four. Multiplying 7.3 by 0.0001 returns 0.00073.

Check the coefficient and exponent

• The coefficient's absolute value is at least 1 and less than 10.
• A positive exponent usually represents a number greater than or equal to 10.
• A negative exponent represents a nonzero number smaller than 1.
• A zero exponent leaves the coefficient unchanged.
• A negative original number keeps a negative coefficient; the exponent records scale, not sign.

Compare or multiply numbers

For positive numbers in normalized scientific notation, compare exponents first; the greater exponent indicates the larger number. If exponents match, compare coefficients. To multiply, multiply coefficients and add exponents, then normalize. For example, (2 × 10³)(3 × 10⁴) = 6 × 10⁷.

Count decimal moves with place value in mind

For a positive number greater than or equal to 10, move the decimal left until one nonzero digit remains to its left; the number of places moved is the positive exponent. For 58,200, the decimal moves four places to give 5.82 × 10⁴. For a number between 0 and 1, move the decimal right until the coefficient is at least 1; that creates a negative exponent. The value 0.0000582 is 5.82 × 10⁻⁵ because five places are crossed.

Do not count visible zeroes as the method. Write the original number with an understood decimal point, move that point until the coefficient is normalized, and count each place. Multiplying by 10 to a positive power shifts place value to the right; a negative power represents division by a positive power of 10. Expanding the result back into ordinary notation is the fastest sign check.

Handle zeros, signs, and normalization

Zero cannot be represented as a × 10ⁿ with a nonzero normalized coefficient, because zero remains zero for every exponent. A negative original number keeps its negative sign in the coefficient: −42,000 = −4.2 × 10⁴. The exponent describes scale, not whether the number is positive or negative. In normalized notation, the absolute value of the coefficient is at least 1 and less than 10; 42 × 10³ is equivalent in value but not normalized.

Compare numbers and calculate

When two positive numbers are normalized and have different exponents, the greater exponent identifies the larger magnitude. If exponents match, compare coefficients. For negative numbers, compare absolute values first and then remember that the value closer to zero is greater: −2 × 10³ is greater than −5 × 10³. When multiplying, multiply coefficients and add exponents. When dividing, divide coefficients and subtract exponents. Normalize the coefficient afterward if it falls outside the interval from 1 to 10.

For example, (6 × 10⁵)(4 × 10⁻²) = 24 × 10³, which must be normalized to 2.4 × 10⁴. The value is 24,000. A common mistake is to multiply the exponents instead of adding them, or to forget the final normalization. For addition and subtraction, exponents cannot simply be added: first rewrite both terms with the same power of 10, then combine coefficients.

Estimate before accepting the answer

• A positive exponent of 4 means the coefficient is scaled by 10,000, not by 4.
• A negative exponent makes a nonzero coefficient smaller by a power of ten.
• Equivalent forms such as 0.42 × 10⁶ and 4.2 × 10⁵ have the same value, but only the second is normalized.
• When adding quantities with very different magnitudes, the smaller addend may have little effect, but it is not automatically zero.
• Keep significant digits consistent with the information supplied; do not imply unsupported measurement precision.

Normalize a number without changing its value

Scientific notation writes a nonzero number as a × 10ⁿ, with 1 ≤ |a| < 10. For a large number, move the decimal point left until one nonzero digit remains before it; the number of places moved is a positive exponent. For 52,000,000, write 5.2 × 10⁷. The positive exponent shows a value larger than 1.

For a number between 0 and 1, move the decimal point right until the coefficient is between 1 and 10; the exponent is negative. For 0.00084, move four places right to obtain 8.4 × 10⁻⁴. The negative exponent indicates a small positive number, not a negative value.

Read each digit by its place value

In 6.31 × 10⁴, the coefficient 6.31 is multiplied by 10,000, giving 63,100. The 3 in the coefficient represents 0.3 of 10,000, or 3,000 in the expanded value; the 1 represents 0.01 of 10,000, or 100. Scientific notation makes magnitude compact while preserving place value.

Compare numbers by first comparing powers of ten when coefficients are normalized. A number written 7.1 × 10⁵ is larger than 9.8 × 10⁴ because the first has a higher exponent. If exponents match, compare coefficients: 6.3 × 10³ is greater than 5.9 × 10³. For negative values, compare signs and absolute magnitudes separately.

Convert back and estimate magnitude

To convert scientific notation to standard form, move the decimal point right for a positive exponent and left for a negative exponent. Include zeros so the place value is correct. For 3.06 × 10⁻³, move three places left to get 0.00306. Count the movement and check the magnitude; the result should be less than one.

Scientific notation is useful for very large or very small measurements, but the exponent must match the number of decimal moves. Estimate the order of magnitude before accepting a calculation. If a microscopic length is converted to a value of millions of meters, a sign or exponent likely changed incorrectly.

• Keep the coefficient between 1 and 10 in absolute value.
• Use a positive exponent for large values and a negative exponent for values between 0 and 1.
• Count decimal moves carefully when converting forms.
• Compare exponents first for normalized positive values.
• Estimate the result’s magnitude to check the exponent sign.

Exam takeaway

Normalize the coefficient, count decimal places carefully and preserve the sign of the exponent. Verify by expanding the notation back into standard form.