Standard Form Calculator

A number is in standard form — the same thing as scientific notation — when it is written as

$a \times 10^{n}, \qquad 1 \le |a| < 10, \quad n \in \mathbb{Z}$

The coefficient $a$ carries the significant digits; the exponent $n$ carries the size. $4\,780\,000 = 4.78 \times 10^{6}$ and $0.00062 = 6.2 \times 10^{-4}$.

Watch the vocabulary — it flips between countries. In UK and international syllabuses, standard form is the $a \times 10^n$ version. In US textbooks the same form is called scientific notation, and standard form or standard notation means the plain decimal expansion, 4780000. Two other unrelated uses exist: the standard form of a linear equation, $Ax + By = C$, and of a quadratic, $ax^2 + bx + c = 0$. Read the question to see which is wanted.

Why it earns its place: it keeps significant figures visible and makes very large and very small quantities comparable at a glance — a bacterium at $2 \times 10^{-6}$ m against a red blood cell at $8 \times 10^{-6}$ m is an obvious ratio; 0.000002 against 0.000008 is not.

Ordinary number → standard form

1. Place the decimal point after the first non-zero digit to get $a$.
2. Count how many places it moved: left is a positive exponent, right a negative one.
3. Drop leading zeros; keep the significant digits.

$4\,780\,000$: the point moves 6 places left, so $4.78 \times 10^{6}$. $0.00062$: 4 places right, so $6.2 \times 10^{-4}$.

Standard form → standard notation

Move the decimal point $|n|$ places — right for positive $n$, left for negative — padding with zeros. $3.2 \times 10^{-5} = 0.000032$.

Calculating in standard form

$(a \times 10^{m})(b \times 10^{n}) = ab \times 10^{m+n}, \qquad \frac{a \times 10^{m}}{b \times 10^{n}} = \frac{a}{b} \times 10^{m-n}$

Multiply or divide the coefficients, add or subtract the exponents, then renormalise so that $1 \le |a| < 10$.

Addition and subtraction are different: the exponents must match first. Rewrite the smaller term with the larger exponent, add the coefficients, then renormalise.

• Leaving the coefficient out of range. $47.8 \times 10^{5}$ is the right value but not standard form; shift it to $4.78 \times 10^{6}$.
• Getting the sign of the exponent backwards. Numbers smaller than 1 take a negative exponent. A quick check: $10^{-4}$ should read as $0.0001$, so $6.2 \times 10^{-4} = 0.00062$.
• Adding exponents when adding numbers. $10^{m+n}$ is the rule for multiplication only.
• Miscounting zeros. For $10^{-5}$ there are 4 zeros between the point and the digit 3 in $0.000032$, not 5 — the exponent counts decimal places moved, not zeros written.
• Losing significant figures. $4.78 \times 10^6$ claims 3 significant figures; writing $4.780 \times 10^6$ claims 4 and asserts precision you may not have.
• Trusting calculator shorthand. A display of 4.78E6 or 4.78 06 means $4.78 \times 10^{6}$; it is not an acceptable way to write the answer.