N8, N9 - Exact multiples of pi and standard form

N8, N9 - Exact multiples of pi and standard form

Keep exact values intact and write very large or small numbers compactly. N8 develops exact calculations with fractions and multiples of pi; N9 develops standard form, its four operations and sensible interpretation of calculator displays.

Exact values and multiples of pi

A circle has radius 4.5 cm4.5\,\mathrm{cm}. Is its circumference exactly 28.26 cm28.26\,\mathrm{cm}, exactly 28.27 cm28.27\,\mathrm{cm}, or neither? The distinction matters whenever a question asks for an exact answer.

You already know

Equivalent fractions have the same value. To add fractions, use a common denominator; to simplify a fraction, cancel common factors.

An exact value preserves the exact quantity. Thus 13\frac13 is exact, while 0.3330.333 is an approximation. Three copies of the fraction give exactly one; three copies of that finite decimal give 0.9990.999. A small rounding error can survive a calculation.

Exact multiple of pi

An expression such as 3π3\pi or 5π2\frac{5\pi}{2} keeps the circle constant π\pi exact. Neither 3.143.14 nor 227\frac{22}{7} equals π\pi; both are approximations.

Treat π\pi as a common factor: add the coefficients, or cancel a factor appearing in numerator and denominator. For example, 3π+5π=8π3\pi+5\pi=8\pi, 7π−2π=5π7\pi-2\pi=5\pi, 12π8=3π2\frac{12\pi}{8}=\frac{3\pi}{2} and 15π6π=52\frac{15\pi}{6\pi}=\frac52.

Circle circumference

C=2πr=πdC=2\pi r=\pi d

Here rr is the radius and dd the diameter of a circle.

On the formula sheet (2026 version; the 2027 sheet will have the same content).

Keeping π\pi in the working answers the opening question without guessing a final decimal digit.

Worked example 1: retaining an exact circumference

Question. Find the circumference of a circle of radius 4.5 cm4.5\,\mathrm{cm}, exactly and to two decimal places.

Plan. Substitute into the circumference formula, retaining π\pi until the decimal answer is needed.

C=2π(4.5)substitute the radius=9π cmmultiply the coefficient≈28.27 cmround only at the end\begin{aligned} C&=2\pi(4.5) && \text{substitute the radius} \\ &=9\pi\,\mathrm{cm} && \text{multiply the coefficient} \\ &\approx28.27\,\mathrm{cm} && \text{round only at the end} \end{aligned}

Why? Keeping π\pi avoids replacing the circle constant by a nearby number.

Answer. Exactly 9π cm9\pi\,\mathrm{cm}; approximately 28.27 cm28.27\,\mathrm{cm}.

Check. 9×3.14=28.269\times3.14=28.26 is close, but it is not the exact circumference.

Change only the radius to 0.45 cm0.45\,\mathrm{cm}. The circumference should become ten times smaller.

Worked example 2: changing the scale of the circle

Question. Find the circumference for radius 0.45 cm0.45\,\mathrm{cm}, exactly and to two decimal places.

Plan. Use the same formula and postpone rounding.

C=2π(0.45)substitute=0.9π=9π10 cmkeep an exact coefficient≈2.83 cmround the final value\begin{aligned} C&=2\pi(0.45) && \text{substitute} \\ &=0.9\pi=\frac{9\pi}{10}\,\mathrm{cm} && \text{keep an exact coefficient} \\ &\approx2.83\,\mathrm{cm} && \text{round the final value} \end{aligned}

Why? A fractional coefficient is still an exact multiple of π\pi.

Answer. Exactly 9π10 cm\frac{9\pi}{10}\,\mathrm{cm}; approximately 2.83 cm2.83\,\mathrm{cm}.

Check. The exact answer is one tenth of 9π9\pi, as expected.

The same care works when combining exact lengths: 2π3+5π6=4π+5π6=3π2\frac{2\pi}{3}+\frac{5\pi}{6}=\frac{4\pi+5\pi}{6}=\frac{3\pi}{2}. Use == for equal values and ≈\approx for rounded ones.

Watch out

Wrong: π=3.14\pi=3.14. Right: π≈3.14\pi\approx3.14. An answer containing π\pi is complete when an exact value is requested.

Now change the radius once more and keep both kinds of answer distinct.

Writing large and small numbers in standard form

Standard form separates a number's digits from its scale. For 43 00043\,000, use 4.34.3 multiplied by 10 00010\,000.

Standard form

A positive number in standard form is A×10nA\times10^n, where 1≤A<101\leq A<10 and nn is an integer. The coefficient AA carries the significant digits.

A negative exponent means division by a power of ten, not a negative value. The coefficient condition makes the representation unique.

[DIAGRAM: asset_name: Keeping a standard-form value unchanged; asset_slug: 1ma1_m07_standard_scale; caption: Multiplying the coefficient by ten requires dividing the power-of-ten factor by ten.; description: Three equal-value expressions 0.43 times 10 to the 5, 4.3 times 10 to the 4 and 43 times 10 to the 3 show coefficient changes opposite to exponent changes. Only the middle expression has coefficient at least one and below ten. All three equal 43000.]
Multiplying the coefficient by ten requires dividing the power-of-ten factor by ten.

Worked example 3: a large number

Question. Write 43 00043\,000 in standard form, then convert it back.

Plan. Choose a coefficient between one and ten and count the power-of-ten scale.

43 000=4.3×10 000separate digits and scale=4.3×104four factors of ten4.3×104=43 000multiply back\begin{aligned} 43\,000&=4.3\times10\,000 && \text{separate digits and scale} \\ &=4.3\times10^4 && \text{four factors of ten} \\ 4.3\times10^4&=43\,000 && \text{multiply back} \end{aligned}

Why? 43×10343\times10^3 has the same value, but its coefficient is too large.

Answer. 4.3×1044.3\times10^4.

Check. Four place changes take 4.34.3 to 43 00043\,000.

Change only the size: keep the digits 4.34.3, but use a number below one.

Worked example 4: a small number

Question. Write 0.000430.00043 in standard form, then convert it back.

Plan. Keep the same coefficient and use a reciprocal power of ten.

0.00043=4.310 000identify the scale=4.3×10−4division gives a negative exponent4.3×10−4=0.00043divide back\begin{aligned} 0.00043&=\frac{4.3}{10\,000} && \text{identify the scale} \\ &=4.3\times10^{-4} && \text{division gives a negative exponent} \\ 4.3\times10^{-4}&=0.00043 && \text{divide back} \end{aligned}

Why? The value is positive and below one, so its exponent is negative.

Answer. 4.3×10−44.3\times10^{-4}.

Check. 0.00043×10 000=4.30.00043\times10\,000=4.3.

The same digits can describe numbers a hundred million times apart: 4.3×1044.3×10−4=108\frac{4.3\times10^4}{4.3\times10^{-4}}=10^8. The exponent carries that enormous difference.

Values between one and ten use exponent zero: 6.8=6.8×1006.8=6.8\times10^0. One is an allowed coefficient; ten is not, so 10×103=1×10410\times10^3=1\times10^4. For a negative quantity, put a minus sign before the standard form of its magnitude, such as −4.3×104-4.3\times10^4. Zero has no representation satisfying the coefficient condition; write it normally.

Watch out

0.43×1050.43\times10^5 and 43×10343\times10^3 both equal 43 00043\,000, but neither is standard form. Check value and coefficient separately.

The next conversion changes just the digits in the small number.

The four operations in standard form

The operation decides what happens to the powers. Products multiply coefficients and add exponents; quotients divide coefficients and subtract exponents. Sums and differences first need a common power of ten.

Powers of ten in products and quotients

10a×10b=10a+b,10a÷10b=10a−b10^a\times10^b=10^{a+b},\qquad10^a\div10^b=10^{a-b}

Here aa and bb are integers. These rules apply to products and quotients.

Not on the formula sheet: learn it.

For addition and subtraction, 4.2×105+7×104=(4.2+0.7)×105=4.9×1054.2\times10^5+7\times10^4=(4.2+0.7)\times10^5=4.9\times10^5. Similarly, 6.1×10−3−8×10−4=(6.1−0.8)×10−3=5.3×10−36.1\times10^{-3}-8\times10^{-4}=(6.1-0.8)\times10^{-3}=5.3\times10^{-3}. Matching scales is like finding a common denominator.

Worked example 5: choosing rules for four operations

Question. Let a=3×104a=3\times10^4 and b=8×10−2b=8\times10^{-2}. Work out abab, a/ba/b, a+ba+b and a−ba-b in standard form.

Plan. Use index laws for the product and quotient; rewrite bb on the 10410^4 scale for the sum and difference.

ab=24×102=2.4×103multiply; normaliseab=0.375×106=3.75×105subtract −2 from 4b=0.000008×104match the scalea+b=3.000008×104add coefficientsa−b=2.999992×104subtract coefficients\begin{aligned} ab&=24\times10^2=2.4\times10^3 && \text{multiply; normalise} \\ \frac ab&=0.375\times10^6=3.75\times10^5 && \text{subtract }-2\text{ from }4 \\ b&=0.000008\times10^4 && \text{match the scale} \\ a+b&=3.000008\times10^4 && \text{add coefficients} \\ a-b&=2.999992\times10^4 && \text{subtract coefficients} \end{aligned}

Why? Subtracting a negative exponent gives 4−(−2)=64-(-2)=6; dividing by 0.080.08 increases the number.

Answer. The four results are 2.4×1032.4\times10^3, 3.75×1053.75\times10^5, 3.000008×1043.000008\times10^4 and 2.999992×1042.999992\times10^4.

Check. Ordinary numbers give 30 000×0.08=240030\,000\times0.08=2400 and 30 000÷0.08=375 00030\,000\div0.08=375\,000. Adding or subtracting 0.080.08 barely changes 30 00030\,000.

Change only the exponent of bb, from −2-2 to 44. This makes addition much easier, but the same rules still apply.

Worked example 6: changing one exponent

Question. Now let a=3×104a=3\times10^4 and b=8×104b=8\times10^4. Find the same four results.

Plan. Keep the product and quotient rules; the sum and difference already have matching scales.

ab=24×108=2.4×109multiply; normaliseab=0.375×100=3.75×10−1divide; normalisea+b=11×104=1.1×105add; normalisea−b=−5×104subtract coefficients\begin{aligned} ab&=24\times10^8=2.4\times10^9 && \text{multiply; normalise} \\ \frac ab&=0.375\times10^0=3.75\times10^{-1} && \text{divide; normalise} \\ a+b&=11\times10^4=1.1\times10^5 && \text{add; normalise} \\ a-b&=-5\times10^4 && \text{subtract coefficients} \end{aligned}

Why? The difference is negative because a<ba<b; normalising must not lose that sign.

Answer. 2.4×1092.4\times10^9, 3.75×10−13.75\times10^{-1}, 1.1×1051.1\times10^5 and −5×104-5\times10^4.

Check. 30 000÷80 000=0.37530\,000\div80\,000=0.375 and 30 000−80 000=−50 00030\,000-80\,000=-50\,000.

In a longer calculation keep brackets around each complete standard-form value. For instance, (4×106)(3×10−2)8×103=128×104−3=1.5×101\frac{(4\times10^6)(3\times10^{-2})}{8\times10^3}=\frac{12}{8}\times10^{4-3}=1.5\times10^1. A difference can be zero; leave zero as zero.

Watch out

Wrong: 3×104+8×104=11×1083\times10^4+8\times10^4=11\times10^8. Addition does not add exponents. The correct sum is 11×104=1.1×10511\times10^4=1.1\times10^5.

Change only the coefficient of bb in the matched question.

Scale, calculator displays and units

A calculator display still needs to be read as a quantity, with an exponent and a unit. “6.2E−5” means 6.2×10−5=0.0000626.2\times10^{-5}=0.000062, not 6.2−56.2-5. Use your calculator's exponent-entry key and check a manageable entry first.

For positive standard-form values, compare exponents first, then coefficients if exponents match. Thus 8.7×104<1.2×1058.7\times10^4<1.2\times10^5, while 4.8×10−6>3.9×10−64.8\times10^{-6}>3.9\times10^{-6}. A difference of two exponents suggests hundreds, but coefficients still affect an exact ratio: 6×1072×105=3×102=300\frac{6\times10^7}{2\times10^5}=3\times10^2=300.

Worked example 7: reading a display and converting units

Question. An instrument displays “4.5E−3 mg”. Write the mass as an ordinary number in milligrams, then in grams. Use 1000 mg=1 g1000\,\mathrm{mg}=1\,\mathrm{g}.

Plan. Interpret the display before converting its unit.

m=4.5×10−3=0.0045 mgread the exponent=0.0045÷1000 gconvert milligrams=0.0000045 gdivide by a thousand\begin{aligned} m&=4.5\times10^{-3}=0.0045\,\mathrm{mg} && \text{read the exponent} \\ &=0.0045\div1000\,\mathrm{g} && \text{convert milligrams} \\ &=0.0000045\,\mathrm{g} && \text{divide by a thousand} \end{aligned}

Why? Standard form changes notation; it does not itself change milligrams into grams.

Answer. 0.0045 mg=0.0000045 g0.0045\,\mathrm{mg}=0.0000045\,\mathrm{g}.

Check. 0.0000045×1000=0.00450.0000045\times1000=0.0045.

Change only the display exponent from −3-3 to −2-2. Both masses should become ten times larger.

Worked example 8: checking a display's scale

Question. Convert “4.5E−2 mg” to ordinary numbers in milligrams and grams.

Plan. Use the same two separate steps.

m=4.5×10−2=0.045 mgread the display=0.045÷1000 gconvert units=0.000045 gcalculate\begin{aligned} m&=4.5\times10^{-2}=0.045\,\mathrm{mg} && \text{read the display} \\ &=0.045\div1000\,\mathrm{g} && \text{convert units} \\ &=0.000045\,\mathrm{g} && \text{calculate} \end{aligned}

Why? Increasing the exponent by one multiplies the value by ten, even when both exponents are negative.

Answer. 0.045 mg=0.000045 g0.045\,\mathrm{mg}=0.000045\,\mathrm{g}.

Check. Each answer is ten times its counterpart in Example 7.

Units travel through calculations too: (2.4×10−6 J)×(5×104)=0.12 J(2.4\times10^{-6}\,\mathrm{J})\times(5\times10^4)=0.12\,\mathrm{J} for the total energy of that many pulses. Retain digits until the end: 7.4×1083.1×102≈2.387096774×106\frac{7.4\times10^8}{3.1\times10^2}\approx2.387096774\times10^6, which is 2.39×1062.39\times10^6 to three significant figures. Extra display digits do not make approximate inputs exact.

Watch out

Wrong: “4.5E−2 mg” means 0.045 g0.045\,\mathrm{g}. The display still uses milligrams. Right: it means 0.045 mg=0.000045 g0.045\,\mathrm{mg}=0.000045\,\mathrm{g}. Reading notation and converting units are separate steps.

The next display changes only its coefficient.

Quick Recap

  • Keep fractions and π\pi exact until a decimal approximation is requested.
  • Standard form is A×10nA\times10^n, with 1≤A<101\leq A<10 and integer nn.
  • Products and quotients use index laws; sums and differences need matching scales.
  • Interpret calculator exponents, retain units and round only at the end.

The opening circumference is neither decimal exactly: it is exactly 9π cm9\pi\,\mathrm{cm}, approximately 28.27 cm28.27\,\mathrm{cm}.