Scientific Notation

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.

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?