Math101 · Independent Ontario learning library
Math101learn.math101.caScientific Notation
Scientific notation expresses extremely large or small values as a number from 1 to 10 multiplied by a power of ten.
Scientific notation writes a nonzero number as $a\times10^n$, where $1\le |a|<10$ and $n$ is an integer.
Why this notation exists
Long strings of zeros hide a number’s scale and invite copying errors. The distance $149\,600\,000$ km is easier to compare and calculate with as $1.496\times10^8$ km. A cell measuring $0.000012$ m becomes $1.2\times10^{-5}$ m.
The coefficient $a$ carries the significant digits. The power of ten records place value.
From standard form to scientific notation
Move the decimal point until exactly one nonzero digit remains to its left. Count the moves.
- Moving left produces a positive exponent because the original number is large.
- Moving right produces a negative exponent because the original number is between $-1$ and $1$.
Back to standard form
The exponent tells how the coefficient is scaled. For $n>0$, move the decimal $n$ places right. For $n<0$, move it $|n|$ places left.
Zeros inserted during the move are place holders; they do not change the significant digits.
Multiplication and division
Multiply or divide coefficients, then use exponent rules on the powers of ten:
If the new coefficient is not between $1$ and $10$, normalize it. For example,
Addition and subtraction
The powers of ten must match before coefficients can combine. This is the same idea as needing like units or like terms.
Adding the exponents here would be incorrect; exponent addition belongs to multiplication.
Comparing and estimating
Compare exponents first. A number with exponent $9$ is larger than a positive number with exponent $7$. If exponents match, compare coefficients. Estimation also becomes quick: $(2.1\times10^4)(3.9\times10^2)$ is near $(2\times4)\times10^6=8\times10^6$.
Calculator notation
Many calculators display $3.6\times10^8$ as 3.6E8. The E means “times ten to the power,” not multiplication by a variable named $E$. Use the calculator’s exponent-entry key rather than typing a long row of zeros.
Common mistakes
Allowing a coefficient such as $32$. Rewrite $32\times10^4$ as $3.2\times10^5$.
Choosing the exponent sign from the direction of the final decimal move. Verify by asking whether the original magnitude is large or tiny.
Adding exponents while adding numbers. Match powers first; add only the coefficients.
Dropping units or significant digits. Scientific notation does not remove the measurement context.
Quick self-check
- Is the coefficient at least $1$ but less than $10$ in magnitude?
- Does the exponent reproduce the original size?
- Did I normalize after multiplying or dividing?
- For addition, do the powers of ten match?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Which standard-form number equals 6.3 × 10⁻⁴?
- The exponent is −4.
- Move the decimal in 6.3 four places left.
- 6.3 × 10⁻⁴ = 0.00063.
End of lesson
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