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The answer √2 is exact. The decimal 1.414 is an approximation to it. A result does not become more accurate merely because it contains familiar decimal digits, and an exact surd is not an unfinished calculation when that is the requested answer form.
However, not every decimal is approximate. The number 0.5 equals one half exactly. The important distinction is whether the written representation preserves the intended value or replaces it with a rounded or truncated nearby value. Read the question’s required form before choosing the final presentation.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–20
Sections 21–25
Sections 26–30
The positive number √2 is defined by its square being exactly two. Its decimal expansion begins 1.414213562…, continuing without terminating or repeating. Writing only 1.414 replaces that exact value with a slightly different number.
If a question asks for three decimal places, write √2 ≈ 1.414. If it asks for an exact answer, keep √2 or an equivalent exact expression. If it asks for a simplified surd, carry out the appropriate exact simplification rather than substituting a decimal.
The symbol = asserts equality, while ≈ signals approximation. Those symbols help the reader see where exact reasoning changes into a numerical presentation. The goal is not to avoid decimals, but to use the representation and precision the task actually requests.
Fractions, decimals, roots, measurement and rounding appear in different forms throughout secondary mathematics. This guide offers a reading route across those connections; it does not imply that every surd manipulation shown is required in every G1, G2 or G3 lesson.
The SEAB 2027 G1 Mathematics syllabus and G3 Mathematics syllabus distinguish non-exact numerical presentation from other answer instructions. Specific directions in a question matter, including directions concerning π or a particular degree of accuracy.
Do not replace those instructions with always leave everything exact or always give a decimal. A mathematically valid expression and a correctly presented answer are connected but separate checks. The final line must match the requested form and context.
For broader support, use eduKate’s guide to exact answers, rounding, significant figures and bounds. This article concentrates on why an exact expression is a complete value rather than a less accurate version of a decimal.
The fraction one third identifies one exact rational number. The expression √2 identifies one exact positive irrational number. The expression 3π identifies exactly three times the mathematical constant π. None requires a finite decimal expansion to be a well-defined value.
An exact expression can still need simplification if the question requests a particular form. For example, √8 equals 2√2 exactly. Both name the same value, but 2√2 is the usual simplified surd form because the square factor four has been extracted.
The fact that a calculator can display a decimal approximation does not make the original expression incomplete. It supplies another representation, often useful for comparison or practical reporting, while potentially discarding information through finite precision.
Think of exact notation as naming the number precisely. A decimal approximation locates it within a nearby numerical interval. Both are useful, but they do different jobs and should not be mistaken for identical values when the approximation has changed the number.
The decimal 0.5 is exactly 5/10, which simplifies to 1/2. The decimal 0.125 is exactly 125/1000, or 1/8. A finite decimal written as a mathematical value represents a rational number exactly.
The issue arises when that decimal is used as a replacement for another intended value. If √2 is replaced by 1.414, then 1.414 is itself an exact rational number, but it is only an approximation to √2. Exactness depends on the relationship being claimed.
Likewise, 0.33 is exactly thirty-three hundredths, but it is not exactly one third. Writing one third equals 0.33 would make a false equality. Writing one third is approximately 0.33 correctly describes the rounded comparison.
This distinction avoids the misleading slogan decimals are not exact. Some are exact representations of the intended number; some are rounded reports of a different intended number. Ask what value the problem started with and how the decimal was obtained.
The notation 0.333… with the repeating-three pattern explicitly intended forever represents one third exactly. It is not the same as the finite decimal 0.333, which equals 333/1000 and is slightly smaller than one third.
The dots need a clear mathematical meaning. In an explanation, state that the digit three repeats indefinitely. Merely copying a calculator screen followed by dots does not always identify the intended repeating pattern unambiguously.
For one third, a fraction is usually the shorter exact representation. It avoids writing many digits and makes operations such as multiplying by three immediate: three times one third equals one exactly.
A learner who understands that recurring decimals can be exact is better prepared to distinguish rational values from irrational ones. The difference is not whether a decimal is long, but whether its expansion terminates, eventually repeats, or does neither.
Every finite decimal is rational because it can be written as an integer divided by a power of ten. The number √2 is irrational, meaning it cannot be written as a ratio of two integers with a nonzero denominator.
Therefore no finite decimal equals √2 exactly. Adding more digits can improve an approximation, but it cannot turn a terminating decimal into that irrational value. The exact expression √2 already identifies the number without this limitation.
You do not need to prove irrationality every time you use a surd. The important operational point is that a finite display is a numerical approximation, while the radical expression can retain exactness through later calculations.
This also explains why a very long calculator display is not automatically more exact than a short symbolic answer. Length of notation and mathematical exactness are different properties. A compact expression can carry more precise information than many displayed digits.
| Intended value and representation | Exact for that value? | Useful interpretation |
|---|---|---|
| 1/2 written as 0.5 | Yes | Equivalent terminating decimal |
| 1/3 written as 0.33 | No | Approximation to two decimal places |
| 1/3 written as 0.333… with 3 recurring | Yes | Infinite recurring decimal |
| √2 written as √2 | Yes | Exact principal square root |
| √2 written as 1.414 | No | Approximation to three decimal places |
| √8 written as 2√2 | Yes | Equivalent simplified surd |
| π written as 3.142 | No | Decimal approximation to π |
The first column names the intended value, which is essential to judging the second. The finite number 1.414 is exact as a rational number in its own right; it becomes approximate when used to represent √2.
Use this table to describe a relationship, not to rank all answer formats. A question about measured quantities may appropriately require a decimal, while an exact algebra question may appropriately require a fraction or surd.
The statement √8 = 2√2 is exact. The statement 2√2 ≈ 2.828 is approximate. A chain such as √8 = 2√2 ≈ 2.828 shows precisely where the representation changes.
Writing √8 = 2.828 would assert that the irrational number equals a finite rational number. In an informal answer box the rounding instruction may make the intention understandable, but in explicit working the approximation symbol is clearer.
The same principle applies to fractions: 1/3 ≈ 0.333, not exact equality with that finite decimal. By contrast, 1/8 = 0.125 is an exact equality because both representations name the same rational number.
These symbols are not merely presentation preferences. They distinguish a mathematical identity from a nearby numerical estimate. Using them accurately helps later steps avoid treating a rounded input as though no information had been lost.
Take √50. Since fifty equals twenty-five times two, √50 = √25 × √2 = 5√2. The extracted factor is exact because the factors are nonnegative and twenty-five is a perfect square.
Similarly, √72 = √(36 × 2) = 6√2. These forms reveal that the two surds are like terms: 5√2 + 6√2 = 11√2. Separate rounded decimals would obscure that shared structure.
Do not assume √(a + b) = √a + √b. For example, √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. The product rule for nonnegative factors does not become an addition rule.
For a fuller skills route, read the core aim of surd mastery. Here the central benefit is that exact simplification can make the calculation clearer, not merely preserve an unfamiliar-looking symbol.
Consider three times one third. Exact fraction arithmetic gives 3 × 1/3 = 1. If you replace one third by 0.333 first, multiplying by three gives 0.999. That is the exact product of the rounded input, but not the exact product of the original fraction.
The discrepancy comes from the approximation, not from multiplication behaving unpredictably. Each copy of one third was replaced by a slightly smaller value, so three copies remain slightly short of one.
Keeping fractions until the end avoids that unnecessary loss. The same principle applies to surds and π: exact structure can simplify before a final numerical conversion is needed.
This does not prohibit using decimals when they are exact and convenient. Multiplying three by 0.5 gives 1.5 exactly. The decision depends on whether the chosen decimal preserves the original value, not on a preference for one visual format.
A square has area two square units. If its side length is s, then s² = 2 and s is positive. Therefore s = √2 units. That is a complete exact side length even though it is not a whole number or terminating decimal.
Its perimeter is 4√2 units. Multiplying the exact side by four preserves the exact relationship. A decimal approximation can then be supplied if the question requests a stated accuracy.
If the side is prematurely rounded to 1.41, squaring that value gives 1.9881 rather than two. The rounded side describes a slightly different square. The original model still has exact area two and exact side √2.
The earlier guide to square-root notation and equation solutions provides the sign connection. The positive side-length restriction chooses √2 rather than negative √2, while the issue of decimal approximation is a separate presentation decision.
Suppose a rectangle has exact side lengths 2√3 and 3√3 units. Its area is (2√3)(3√3) = 6 × 3 = 18 square units. The irrational side factors combine to give an integer area.
If the sides are approximated as 3.464 and 5.196 first, their product is 17.998944. That is close to eighteen, but the small discrepancy is unnecessary. Exact multiplication supplied the simple result directly.
This example is a useful answer to the concern that surds are unfinished numbers. They can preserve algebraic relationships that are harder to see after conversion to decimals. Exact notation sometimes reduces computation rather than adding complexity.
Check that the original side lengths are mathematical exact values. If they were rounded measurements instead, writing an exact symbolic expression would not make the physical rectangle’s dimensions perfectly known. Exact calculation and measurement uncertainty are different layers of the problem.
Suppose an exact intermediate length is √2 and a later calculation multiplies it by one hundred. The exact result is 100√2, approximately 141.421356. If √2 was rounded to 1.41 first, the later result would be 141.
The intermediate error of about 0.0042 has been multiplied by one hundred. This illustrates why early rounding can matter even when the initial difference seems small. Later operations can magnify or otherwise transform that difference.
Use exact expressions or the calculator’s retained internal value through the working where appropriate. Convert to the required final precision at the end, unless the question explicitly instructs a rounding step earlier.
Do not confuse retaining a stored value with retyping only the displayed digits. Depending on the calculator, those actions can use different precision. The mathematical aim is to avoid deliberately replacing an available accurate intermediate value with an unnecessarily shortened one.
Question: a rectangle has exact side lengths 3√2 centimetres and 2√2 centimetres. Find its perimeter in simplified exact form, then give an approximation to three significant figures.
The perimeter is twice the sum of adjacent sides: P = 2(3√2 + 2√2) = 2(5√2) = 10√2 centimetres. This is the exact answer. The common radical allows the side terms to combine before any decimal calculation.
Numerically, 10√2 is approximately 14.14213562. To three significant figures, the first three significant digits are one, four and one, and the next digit is four, so the rounded perimeter is 14.1 centimetres.
Write P = 10√2 cm ≈ 14.1 cm to three significant figures. The exact expression and decimal answer are both useful, but they serve the two separate requests. Do not replace the exact answer with the decimal when both were asked for.
Read each instruction before calculating. Keep exact arithmetic until a numerical approximation is requested. All listed lengths in these mathematical examples are exact, not measured records, unless explicitly described otherwise.
Question A. Express one eighth as an exact decimal. Question B. Explain whether 0.33 equals one third exactly. Question C. Simplify √32 exactly. Question D. Find 3√2 + 5√2 in exact form.
Question E. A square has exact side √5 units. Find its area and perimeter exactly. Question F. A rectangle has exact sides √7 and 2√7 units. Find its exact area. Question G. Give √2 to two decimal places and use an appropriate equality or approximation symbol.
Question H. Calculate three times one third exactly, then compare with three times 0.333. Question I. Give 10√2 to three significant figures. Question J. Decide whether √9 + √16 equals √25 and explain.
Question K. A question explicitly says use π = 3.142. Should you silently replace that instruction with an exact answer involving π? Question L. A length is recorded as 1.4 metres after rounding. Does writing 7/5 metres prove the original physical length was known exactly?
Answer A. One eighth equals 0.125 exactly. Answer B. No. The finite decimal 0.33 is thirty-three hundredths, whereas one third has a recurring decimal expansion. We can write 1/3 ≈ 0.33 to two decimal places.
Answer C. √32 = √(16 × 2) = 4√2. Answer D. 3√2 + 5√2 = 8√2. These are like radical terms; their coefficients add while the common √2 remains.
Answer E. The area is (√5)² = 5 square units and the perimeter is 4√5 units. Answer F. The area is √7 × 2√7 = 14 square units. The exact radical factors simplify without needing decimal side lengths.
Answer G. √2 ≈ 1.41 to two decimal places. Answer H. Three times one third equals one, while three times 0.333 equals 0.999. The second calculation uses an approximated version of the original fraction.
Answer I. 10√2 ≈ 14.1 to three significant figures. Answer J. No. The left side is three plus four, or seven; √25 is five. Square roots do not generally distribute over addition.
Answer K. No. Follow the question’s explicit instruction and present the requested numerical result. Exactness does not authorise ignoring the specified method or form. Answer L. No. Seven fifths exactly equals the recorded number 1.4, but it does not remove the uncertainty in the underlying measurement.
SECTION 17 OF 32
16. Decimal places and significant figures answer different requests
Two decimal places means two digits after the decimal point. For √2, that gives 1.41. Three significant figures also gives 1.41 in this particular case, but that agreement does not make the instructions identical.
For a value such as 0.014142…, two decimal places gives 0.01, while three significant figures gives 0.0141. Significant figures count meaningful digits beginning at the first nonzero digit; decimal places refer to fixed positions after the point.
An exact expression does not require choosing one of these precisions until the task asks for an approximation. When a numerical answer is requested, read which convention applies rather than copying the number of digits from an earlier example.
The connected guide to decimal places and significant figures develops this distinction. Keeping exact working and choosing final presentation are complementary habits, not competing methods.
If a question asks for a length correct to one decimal place, writing only √2 does not provide the requested rounded presentation. The expression is exact, but the instruction asks for its numerical value at a stated precision.
Conversely, if the question asks for an exact surd, writing only 1.4 does not meet that request even when it is a correct one-decimal-place approximation. The numerical closeness does not compensate for the missing exact form.
Some questions specify a numerical value for π, while others ask for an answer in terms of π. Follow the supplied direction. Different mathematical tasks can legitimately use different final representations of a related calculation.
Check answer form as part of the task, not as a cosmetic step added after everything else. A complete solution has a valid model, correct calculation, allowed values, appropriate units and the requested presentation.
In a mathematical problem, a side length stated as exactly √2 units supports exact deductions such as area two square units. In a real measurement, a recorded side of 1.4 metres may represent a range of possible actual lengths.
Rewriting 1.4 as 7/5 preserves the recorded number exactly. It does not prove that the physical object has length exactly seven fifths of a metre. A change in notation cannot recover information lost during measurement or rounding.
Similarly, an exact formula applied to uncertain inputs gives an exact expression in terms of those inputs, not perfect knowledge of the physical quantity. The calculation and the measurement model need to be interpreted together.
For connected reading, the measurement and bounds guide explains what rounded records imply. Keep the exactness of a mathematical representation separate from the precision of the data it represents.
Two expressions can produce the same rounded calculator display without being exactly equal. For example, 1.414 and √2 both round to 1.41 to two decimal places, but they are different numbers.
Matching a finite display is therefore evidence of numerical closeness at that precision, not a general proof of identity. To establish exact equality, use a valid symbolic transformation or another exact argument that fits the expressions involved.
For √8 and 2√2, factoring the radicand establishes equality. A decimal check can support error detection, but it is not the reason those expressions are identical. The exact structure provides the proof.
This distinction is especially useful when a calculator displays many digits. More matching digits can make a numerical comparison stronger, but a finite coincidence still does not by itself establish a universal algebraic identity.
Consider the exact expression 100(√101 − 10). The two terms inside the bracket are close, so a very short decimal approximation to √101 can distort their small difference.
Rationalising the difference gives √101 − 10 = (101 − 100)/(√101 + 10) = 1/(√101 + 10). Therefore the original expression equals 100/(√101 + 10) exactly.
The exact transformed expression makes the small positive value clear and avoids subtracting two nearly equal shortened decimals. It is approximately 4.98756, but the algebraic equality does not depend on that display.
This is an extension for learners ready for the manipulation, not a requirement for every subject level. Its purpose is to show another practical benefit of exact structure: algebra can preserve a meaningful difference that coarse intermediate rounding would conceal.
Suppose √2 could be written as p/q, where p and q are positive integers with no common factor. Squaring gives p² = 2q², so p² is even. An odd integer has an odd square, so p must be even.
Write p = 2k. Substituting gives 4k² = 2q², hence q² = 2k². The same reasoning makes q even. Thus p and q share a factor of two, contradicting the assumption that the fraction was in lowest terms.
Therefore √2 cannot be a rational fraction. Since every finite decimal is rational, no finite decimal equals it exactly. This explains why adding digits improves an approximation without replacing the exact radical by a terminating decimal.
The proof is optional connected reading. You can correctly preserve √2 in a calculation before learning a formal irrationality argument. The important everyday habit remains recognising which representation is exact and which is an approximation to it.
Keeping an exact answer does not mean abandoning estimation. Since one squared is one and two squared is four, √2 lies between one and two. Thus 10√2 lies between ten and twenty.
A calculated decimal such as 141.4 for 10√2 would fail that rough check immediately. Exact symbolic working and approximate magnitude checks can support one another rather than being competing approaches.
You can refine the bracket if useful: 1.4² = 1.96 and 1.5² = 2.25, so √2 lies between 1.4 and 1.5. This supports an approximate decimal around 1.41 without claiming that either endpoint is the exact root.
The earlier guide to estimates that reject but do not prove answers explains the boundary of this check. A plausible scale can expose errors, but it does not replace the exact calculation requested.
If a student correctly obtains 4√2 and then writes 5.66 when an exact answer was requested, the algebra may be sound. The first issue is interpreting the final answer form. Preserve the correct simplification while repairing the presentation choice.
If the student writes √32 = √16 + √2, the error is algebraic structure, not rounding. The radicand thirty-two was treated as though multiplication and addition had the same square-root rule. A numerical approximation does not repair that invalid equality.
If the student rounds √2 early and later obtains a slightly different final decimal, inspect the intermediate precision. The method may be correct but using an unnecessarily shortened value. Explain where approximation entered and how later operations affected the error.
These diagnoses suggest different practice. Match the repair to the first wrong step: exact simplification, notation, rounding instructions or retained precision. More calculator work is not automatically the answer to every disagreement involving decimals.
Choose one value, such as √8. Write three valid descriptions: exact unsimplified radical √8, exact simplified radical 2√2, and a decimal approximation to a stated precision. Mark exact equalities and the approximation boundary clearly.
Next perform one operation before and after rounding. For example, square the exact √2 and compare with squaring 1.41. Explain why the results differ without calling either multiplication calculation mysterious or incorrect.
Then change only the instruction: exact form, two decimal places, or three significant figures. Keep the underlying number fixed. This isolates the presentation decision from the calculation and helps a learner see why different final lines can be appropriate for different requests.
Finish by checking whether the inputs were exact mathematical values or rounded measurements. That final distinction prevents symbolic notation from being mistaken for greater physical knowledge than the data actually provide.
First read the required form. Identify which inputs and intermediate values are exact, and which are measurements or approximations. Keep fractions, surds or other exact expressions where they preserve useful structure.
Simplify validly before converting unnecessarily. Use = for exact relationships and ≈ when a rounded decimal replaces the intended exact value. If using a calculator, retain enough intermediate precision for the later calculation.
At the final step, apply the requested decimal places, significant figures or exact form. Include the correct units and check that the size is sensible. An exact expression is complete when it matches the task; a decimal is appropriate when the task calls for it.
The mathematics learning hub connects surds, fractions, accuracy, measurement and solution checking. Choose the next route according to the distinction that needs repair rather than treating all numerical-format errors as one difficulty.
The expression √9 equals three exactly, and √(9/16) equals three quarters exactly. Both use a radical symbol, but their values are rational. A symbol’s appearance alone does not classify the resulting number.
By contrast, √2 is irrational. The difference comes from the value under the root and its mathematical properties, not from whether a square-root sign is printed on the page. Simplification can reveal that distinction.
For a nonnegative integer radicand, a perfect square has an integer square root. A positive integer that is not a perfect square has an irrational square root. For rational radicands, analyse the actual fraction rather than assuming the integer-only wording covers every case.
This prevents another misleading shortcut: exact answer means irrational answer. Exact answers can be integers, terminating decimals, fractions, recurring decimals, surds or expressions involving constants, depending on the intended value and requested form.
When studying independently, sort examples by value rather than notation. The pair √9 and 3 belongs together as exact equivalents. The pair √2 and 1.414 belongs in an exact-versus-approximate comparison, even though both are useful representations in suitable tasks.
The expressions 1/√2 and √2/2 are exactly equal. Multiplying numerator and denominator by √2 gives √2/(√2 × √2) = √2/2. No decimal approximation is required at any step.
The denominator is nonzero because √2 is positive. Naming that condition makes the transformation a legitimate multiplication by one rather than a visual rule about moving symbols. For variable expressions, nonzero restrictions become even more important.
If a question asks for a rationalised denominator, √2/2 is the appropriate final form here. If it merely asks for an exact value, both expressions identify the same number, though one may be more convenient for later manipulation.
Do not confuse preferred form with greater accuracy. Rationalising this denominator changes the representation, not the value. Similarly, simplifying √8 to 2√2 does not make the original radical approximately correct; the two are exactly equal.
The reading question is therefore twofold: does the expression preserve the value, and does it satisfy the requested form? Both matter, but they should not be collapsed into a vague judgement that unfamiliar notation is less finished or less accurate.
SECTION 29 OF 32
28. A rounded report identifies a neighbourhood, not the original exact value
When √2 is reported as 1.414 to three decimal places, the reported value is near the original number. Under ordinary rounding for these positive values, numbers from 1.4135 up to but not including 1.4145 share that three-decimal-place report.
Thus a display of 1.414 alone does not uniquely identify √2. Many different exact values round to the same finite decimal. The original expression supplies information that the rounded report no longer retains.
The rational number 1.4141 also rounds to 1.414, yet it is not equal to √2. Matching the report is consistent with numerical closeness, not proof that the underlying values are identical.
This is a practical way to explain information loss without implying that rounding is a mistake. Rounding deliberately reports a value at a chosen resolution. It is appropriate when that resolution matches the task, but it should not be reversed as though it recovered a unique original number.
If later working needs the original value, retain √2 or its stored high-precision approximation rather than reconstructing it from the three-decimal report. That preserves the information already available before the reporting step.
In an invented model with six equally likely labelled outcomes, one selected outcome has probability exactly one sixth. The decimal 0.167 is an approximation to that model probability, not a more complete version of it.
The exact fraction describes the assumed mathematical model. It does not prove that every physical object intended to realise those outcomes is perfectly balanced. The assumptions about equal likelihood belong to the model and require their own justification in a real investigation.
This parallels the distinction between exact geometry and measurements. An exact calculation can follow from stated ideal conditions while real data retain uncertainty. Symbolic notation should not be used to claim greater empirical certainty than the situation supports.
If the question asks for probability to three decimal places, report approximately 0.167. If it asks for an exact fraction, report one sixth. Both presentations can be appropriate, provided their relationship to the model and the requested accuracy is clear.
This final transfer example connects exactness beyond surds. The same principle applies whenever a rounded decimal is being compared with an exact fraction, ratio or expression: identify the intended value, retain its conditions and report the form the question requests.
An exact length of √2 metres and its approximation of 1.414 metres describe the same intended quantity at different numerical precision. Converting the exact length to centimetres gives 100√2 centimetres. The conversion factor changes the numerical scale but does not require rounding.
If a final answer is requested in centimetres to one decimal place, calculate from the retained value and report approximately 141.4 centimetres. Starting from the shortened 1.41 metres instead would produce 141.0 centimetres and lose precision unnecessarily.
The unit change and the approximation are separate operations. Write them as separate steps so the reader can see which equality is exact and where the rounded report begins. This is especially useful when a solution includes more than one conversion.
Do not add precision merely because smaller units produce more digits. A physical measurement originally recorded to limited accuracy does not gain new information by conversion. The exact mathematical conversion preserves the available value; it does not improve the measuring instrument.
This final distinction links the guide back to practical numeracy. Accurate communication requires the value, unit, reference assumptions and requested precision to remain consistent, whether the number is written as a fraction, surd or decimal.
These answers separate the appearance of a number from its relationship to the intended value. Always read the question’s explicit instruction before deciding whether an exact expression or rounded decimal belongs in the final line.
Is a decimal always less accurate than a fraction?
No. The decimal 0.5 and fraction 1/2 are exactly equal. The finite decimal 0.33 is only an approximation to 1/3. The relevant question is whether the chosen representation preserves the intended value.
Is √2 an unfinished answer?
No. It identifies an exact number. If the question asks for an exact answer, it can be complete. If the question asks for a decimal at a stated accuracy, add the requested approximation rather than assuming exact notation overrides the instruction.
Do more calculator digits make an irrational number exact?
No finite terminating decimal equals an irrational number exactly. More correct digits improve the approximation. The exact expression already identifies the intended value without needing its entire nonterminating, nonrecurring decimal expansion to be written out.
What should I remember first?
Exactness concerns the value being preserved, not whether the notation looks familiar. Keep exact structure through the calculation where useful, mark approximation honestly, and give the final form and precision the question requests.