Scientific notation and operations with numbers

To convert an ordinary non-zero number to scientific notation, move the decimal point until one non-zero digit remains before it. Count the moves. If the original number is large, the exponent is positive; if the original number lies between 0 and 1, the exponent is negative.

A reliable sign check is to ask what the power of ten must do. In $7.43\times10^6$, the power must enlarge $7.43$, so the exponent is positive. In $5.21\times10^{-5}$, it must shrink $5.21$, so the exponent is negative.

To convert back to ordinary notation, reverse the process. A positive exponent moves the decimal point right; a negative exponent moves it left. Add placeholder zeros only when necessary, and keep any leading minus sign unchanged.

Scientific notation separates each calculation into a coefficient part and a power-of-ten part. Multiply or divide the coefficients normally, apply the exponent laws to the powers of ten, and normalize only after those steps.

When a scientific-notation expression is raised to a power, the power applies to both the coefficient and the power of ten. Parentheses matter: $(a\times10^m)^p=a^p\times10^{mp}$.

If the calculated coefficient is at least 10, move its decimal left and increase the exponent. If it is between 0 and 1, move its decimal right and decrease the exponent. This normalization step preserves the value.

Addition and subtraction are different from multiplication and division: the powers of ten must match before the coefficients can be combined. This is the same place-value principle that prevents us from directly adding metres to kilometres without converting units.

Choose one exponent, rewrite every term using that common power of ten, then add or subtract the coefficients. Finally normalize and round only if the question requests it.

When two close values are subtracted, many leading significant digits may cancel. Keep extra digits during the calculation because early rounding can create a large relative error in the small difference.

Significant figures describe precision. Start counting at the first non-zero digit. Zeros between non-zero digits are significant, and trailing zeros to the right of a decimal point are significant. Leading zeros only locate the decimal point and are not significant.

To round to $n$ significant figures, keep the first $n$ significant digits and inspect the next digit. If it is 5 or more, increase the last kept digit by 1; otherwise leave it unchanged. Replace discarded place-value digits with zeros when ordinary notation requires them.

Scientific notation makes precision explicit. The coefficient $4.10$ has three significant figures, whereas $4.1$ has two. Do not delete a trailing zero when it communicates requested accuracy.

For multiplication and division, a final answer is often reported to the same number of significant figures as the least precise given measurement, unless the question specifies an accuracy. For addition and subtraction, decimal-place precision may be the more meaningful guide. In IB questions, follow the stated instruction first.

On most scientific calculators and GDCs, use the $\mathrm{EXP}$, $\mathrm{EE}$, or $\times10^x$ key to enter a power of ten. To enter $6.02\times10^{23}$, type $6.02$, press the exponent-entry key, then type $23$. Use the dedicated negative-sign key for a negative exponent if your calculator distinguishes it from subtraction.

Calculator output such as $9.9932\text{E}{-4}$ means $9.9932\times10^{-4}$. Do not multiply by another $10^{-4}$. Set the display to scientific mode only when useful; the value is unchanged when the display mode changes.

Use parentheses around an entire numerator, denominator or powered scientific-notation factor. Store intermediate values or keep the full display instead of copying rounded decimals into later steps.

Estimate before or after calculating. Round coefficients to one convenient significant digit and apply the exponent rules mentally. Your estimate should identify the sign, approximate coefficient and order of magnitude. A calculator answer many powers of ten away from the estimate usually signals an entry or exponent error.

IB questions often embed scientific notation inside geometry, probability, physics, finance or modelling. Translate the context into an expression first, substitute values with units, calculate without premature rounding, then interpret the result in the requested form.

A strong written solution normally contains: the formula or expression used; substitution in consistent units; an intermediate line showing exponent arithmetic when relevant; a normalized answer; the required accuracy; and a contextual unit or sentence.

For a multi-part question, store earlier results at full precision. If a later part says 'hence', use the stated earlier value appropriately but avoid retyping a heavily rounded version when the calculator can retain more digits.

Try each question before opening the worked reasoning mentally. The set progresses from direct conversions to IB-style multi-step calculations. In an exam, write enough method to earn marks even if the final arithmetic goes wrong.