Scientific Notation Converter

A number is in scientific notation when it is written as

$a \times 10^{n}, \qquad 1 \le |a| < 10, \; n \text{ an integer}$

The coefficient $a$ must have exactly one non-zero digit before the decimal point. That single condition is what makes the form unique: $47{,}000$ can be written as $4.7 \times 10^4$ and no other way, so two numbers can be compared just by looking at their exponents.

$4.7 \times 10^4 \;\checkmark \qquad 47 \times 10^3 \;\times \qquad 0.47 \times 10^5 \;\times$

The last two are equal in value but not in form; a question asking for scientific notation will mark them wrong.

The exponent counts places, not zeros. A positive $n$ means a large number, a negative $n$ a small one — never a negative number. $-4.7 \times 10^{4}$ is negative; $4.7 \times 10^{-4}$ is small.

To scientific notation: put the decimal point after the first non-zero digit and count how many places it moved. Moved left (the number was big) → positive exponent. Moved right (the number was small) → negative exponent. So $0.00047 \to 4.7 \times 10^{-4}$, four places right.

Back to standard notation: move the decimal point $n$ places right for positive $n$, left for negative $n$, padding with zeros.

Reading E: calculators and spreadsheets print $6.02 \times 10^{23}$ as 6.02E23. The E means "times ten to the", nothing else.

Multiply and divide: handle the coefficients and the powers separately, using $10^a \times 10^b = 10^{a+b}$ and $10^a \div 10^b = 10^{a-b}$. If the new coefficient falls outside $[1, 10)$, renormalise.

Add and subtract: the exponents must match first. Rewrite the smaller number with the larger exponent, then add the coefficients.

• Leaving the coefficient out of range. $12 \times 10^3$ is not scientific notation; renormalise to $1.2 \times 10^4$.
• Counting zeros instead of decimal places. In $0.00047$ the point moves 4 places, even though there are only 3 zeros after it.
• Adding the exponents when adding numbers. Exponents add only for multiplication. For a sum, match the exponents and add the coefficients.
• Flipping the sign of the exponent. A number smaller than 1 always gets a negative exponent. Sanity check: $10^{-4} = 0.0001$.
• Renormalising without fixing the exponent. Turning $12 \times 10^3$ into $1.2$ means the power must rise to $10^4$; the two changes always move in opposite directions.
• Rounding the coefficient too early in a multi-step calculation, which can shift the final significant digits.