The Central Idea
An exact solution represents a value without numerical approximation.
An approximate solution represents a value by a nearby number.
For example,
\[ \boxed{x=\sqrt{2}} \]is exact, while
\[ \boxed{x\approx1.414} \]is approximate.
The symbol \(=\) asserts equality. The symbol \(\approx\) indicates that two values are close but not necessarily identical.
Exact Values
An exact value preserves the full mathematical information specified by an expression.
Examples include
\[ 7, \qquad \frac{2}{3}, \qquad \sqrt{5}, \qquad \pi, \qquad e, \qquad \frac{-b\pm\sqrt{b^2-4ac}}{2a}. \]These expressions do not need to be integers or terminating decimals to be exact.
Approximate Values
An approximate value replaces an exact value with a nearby numerical representation.
For example,
\[ \pi\approx3.14159 \]and
\[ \sqrt{5}\approx2.23607. \]The decimals are useful approximations, but neither finite decimal is equal to the exact value it represents.
Exact Does Not Mean Simple
Exactness is about preserving value, not about making an expression short or easy to evaluate mentally.
For example,
\[ \frac{7+\sqrt{13}}{4} \]is exact even though it contains a radical.
The decimal
\[ 2.651387\ldots \]may be easier to use numerically, but it is an approximation unless every omitted digit is somehow represented.
A Terminating Decimal Can Be Exact
Decimal notation is not automatically approximate.
For example,
\[ 0.25=\frac14. \]Because \(0.25\) terminates and represents \(\frac14\) exactly, it is an exact value.
Likewise,
\[ 1.375=\frac{11}{8}. \]A decimal becomes approximate when digits have been rounded or truncated from a value that requires additional digits.
Repeating Decimals Can Also Be Exact
A repeating decimal written with its repeating pattern indicated can be exact.
\[ \frac13=0.\overline{3}. \]Here, \(0.\overline{3}\) means that the digit \(3\) repeats forever.
By contrast,
\[ \frac13\approx0.333 \]because the finite decimal \(0.333\) stops after three decimal places.
Why Approximate Solutions Are Necessary
Approximation is not merely a shortcut. It is often unavoidable or operationally preferable.
Approximate solutions arise when:
- an exact value has a nonterminating decimal representation;
- an equation has no convenient closed-form solution;
- measurements are known only to limited precision;
- data contain uncertainty;
- a numerical algorithm produces successive estimates;
- a practical decision requires only a specified tolerance.
Exact Algebraic Solution
Consider
\[ 3x-7=11. \]Add \(7\):
\[ 3x=18. \]Divide by \(3\):
\[ \boxed{x=6}. \]This solution is exact.
Exact Solution Containing a Radical
Consider
\[ x^2=7. \]Taking square roots gives
\[ \boxed{x=\pm\sqrt7}. \]These are exact solutions.
Numerically,
\[ x\approx\pm2.64575. \]The decimal form is approximate.
Exact and Approximate Forms Can Coexist
A solution can be reported in both exact and approximate form.
For example,
\[ x=\frac{5+\sqrt{17}}{2} \]is exact, while
\[ x\approx4.56155 \]is a numerical approximation of the same solution.
The exact form is preferable for symbolic reasoning. The approximate form is often preferable for measurement, graphing, comparison, or computation.
When an Exact Formula Is Available
Some equation families have established exact formulas.
A quadratic equation
\[ ax^2+bx+c=0, \qquad a\ne0, \]has solutions
\[ \boxed{ x= \frac{-b\pm\sqrt{b^2-4ac}}{2a} }. \]Even if the radical produces an irrational number, the formula gives an exact symbolic representation.
When No Convenient Exact Formula Is Available
Consider the equation
\[ x^3+x-1=0. \]This equation has an exact solution in principle, but the exact radical expression is much less useful than a numerical approximation for most elementary applications.
Numerically, the real solution is approximately
\[ \boxed{x\approx0.6823}. \]Numerical methods are designed to produce and control approximations such as this one.
Some Equations Require Numerical Methods
Many important equations combine functions in ways that cannot generally be solved by elementary algebraic rearrangement.
For example,
\[ x=\cos x \]has a real solution near
\[ \boxed{x\approx0.739085}. \]There is no ordinary sequence of inverse algebraic operations that isolates \(x\) in elementary closed form.
Iteration and other numerical methods provide a practical solution.
Equality and Approximation Symbols
The symbols \(=\) and \(\approx\) communicate different mathematical claims.
\[ \sqrt2\ne1.414 \]but
\[ \sqrt2\approx1.414. \]Writing
\[ \sqrt2=1.414 \]would be false because the finite decimal does not contain the complete value of \(\sqrt2\).
Rounding
Rounding replaces a number with a nearby value at a chosen place value.
Suppose
\[ x=7.283641\ldots \]Rounded to the nearest hundredth,
\[ \boxed{x\approx7.28}. \]Rounded to the nearest thousandth,
\[ \boxed{x\approx7.284}. \]Truncation
Truncation removes digits after a selected place without using the next digit to round.
If
\[ x=7.283641\ldots, \]truncating to three decimal places gives
\[ \boxed{x\approx7.283}. \]Rounding to three decimal places instead gives
\[ \boxed{x\approx7.284}. \]Decimal Places
The number of decimal places counts digits to the right of the decimal point.
For example,
\[ 3.1416 \]is written to four decimal places.
If it represents \(\pi\), then
\[ \pi\approx3.1416. \]Significant Figures
Significant figures count meaningful digits beginning with the first nonzero digit.
For example,
\[ 0.004827 \]has four significant figures:
\[ 4,\quad8,\quad2,\quad7. \]To three significant figures,
\[ \boxed{0.004827\approx0.00483}. \]Approximation Depends on the Required Precision
There is no single decimal approximation that is always the correct one.
For \(\sqrt2\),
\[ \sqrt2\approx1.4 \]may be sufficient for a rough estimate,
\[ \sqrt2\approx1.414 \]may be sufficient for ordinary computation, and
\[ \sqrt2\approx1.414213562 \]may be appropriate when greater precision is required.
Approximation Error
If the exact value is \(x\) and an approximation is \(a\), then the difference
\[ x-a \]is the signed error.
Because error is often discussed by magnitude, we commonly use
\[ \boxed{|x-a|}. \]This is the absolute error.
Example: Absolute Error
Suppose
\[ \sqrt2\approx1.41. \]The absolute error is
\[ |\sqrt2-1.41|. \]Numerically,
\[ |\sqrt2-1.41| \approx0.00421356. \]The approximation differs from the exact value by a little more than four thousandths.
Relative Error
Absolute error measures the raw difference. Relative error compares that difference with the size of the exact value.
For \(x\ne0\),
\[ \boxed{ \text{relative error} = \frac{|x-a|}{|x|} }. \]Relative error is useful when the scale of the quantity matters.
Percent Error
Relative error can be expressed as a percentage:
\[ \boxed{ \text{percent error} = \frac{|x-a|}{|x|}\cdot100\% }. \]Percent error is especially common when comparing measurements, predictions, estimates, and observed values.
An Error Bound
Sometimes we do not know the exact error, but we can guarantee that the error is smaller than a specified amount.
If
\[ |x-a|<0.001, \]then \(a\) approximates \(x\) with absolute error less than \(0.001\).
Such a statement gives an error bound.
Tolerance
A tolerance is an acceptable error threshold.
For example, a numerical method may be instructed to continue until
\[ |x-a|<10^{-6}. \]In practical algorithms, the exact value \(x\) may be unknown, so a stopping rule often uses the change between successive approximations or another computable error estimate.
Intervals Can Express Approximate Knowledge
Instead of reporting a single decimal, we can sometimes trap an unknown solution inside an interval.
Suppose we know
\[ 1.414Bracketing methods use this idea systematically.
Approximating a Root by an Interval
Consider
\[ x^2=2. \]Since
\[ 1.4^2=1.96<2 \]and
\[ 1.5^2=2.25>2, \]the positive root satisfies
\[ \boxed{1.4<\sqrt2<1.5}. \]Refining the interval gives a more accurate approximation.
Refining an Approximation
Test the next decimal place:
\[ 1.41^2=1.9881<2 \]and
\[ 1.42^2=2.0164>2. \]Therefore,
\[ \boxed{1.41<\sqrt2<1.42}. \]The interval has narrowed from width \(0.1\) to width \(0.01\).
Successive Approximation
A successive approximation process produces a sequence of increasingly useful estimates.
For \(\sqrt2\), one possible progression is
\[ 1.4,\qquad 1.41,\qquad 1.414,\qquad 1.4142,\qquad 1.41421. \]Each new approximation contains more information about the target value.
Later lessons formalize this idea using numerical tables, iteration, bracketing, bisection, fixed-point methods, Newton's method, and secant methods.
Exact Input Can Produce an Approximate Output
Even when the original problem uses exact quantities, a numerical answer may be reported approximately.
For example, a circle with exact radius
\[ r=5 \]has exact circumference
\[ C=2\pi r=10\pi. \]Numerically,
\[ C\approx31.416. \]The approximation appears only when \(\pi\) is replaced by a finite decimal.
Approximate Input Limits the Meaning of Exact Algebra
Suppose a measured length is reported as
\[ L=12.4\text{ cm}. \]The written number may already represent a rounded measurement rather than a mathematically exact physical length.
Algebra performed with the number \(12.4\) can be symbolically exact, but the physical conclusion cannot contain more reliable information than the measurement supports.
Numerical precision and measurement precision are therefore related but distinct ideas.
Calculator Displays Are Finite
A calculator can display only finitely many digits.
If a calculator displays
\[ \sqrt3\approx1.732050808, \]the displayed decimal is still an approximation.
The calculator may internally carry more digits than it displays, but no ordinary finite display contains the entire nonterminating decimal expansion of \(\sqrt3\).
Stored Precision and Display Precision
A computational tool may store more digits than it shows.
For this reason, it is usually better to keep full calculator precision during intermediate calculations and round only the final reported result.
Repeatedly rounding intermediate results can increase accumulated numerical error.
Example: Rounding Too Early
Suppose a calculation requires
\[ \sqrt2\cdot\sqrt2. \]Exactly,
\[ \boxed{\sqrt2\cdot\sqrt2=2}. \]If we first round
\[ \sqrt2\approx1.41, \]then
\[ 1.41^2=1.9881. \]The early approximation has introduced error into a calculation whose exact value is simply \(2\).
Keep Exact Form as Long as Practical
A useful general strategy is:
- solve symbolically when an exact form is available and manageable;
- preserve exact constants and fractions during intermediate work;
- convert to decimals when a numerical result is needed;
- round according to the required precision.
This strategy reduces unnecessary numerical error.
Approximation in Graphing
Graphs often reveal approximate solutions as intersection points.
If
\[ f(x)=g(x), \]then solutions occur where the graphs of \(y=f(x)\) and \(y=g(x)\) intersect.
A graph may suggest that a solution is near
\[ x\approx2.3, \]but graphical resolution usually limits the number of reliable digits that can be read directly.
Approximation in Tables
Numerical tables can narrow the location of a solution.
Suppose
\[ f(2.3)<0 \]and
\[ f(2.4)>0. \]Under appropriate continuity conditions, this sign change can indicate a root between \(2.3\) and \(2.4\).
The next lesson develops numerical tables and successive approximation in greater detail.
Approximation in Algorithms
Numerical algorithms generally produce approximations rather than exact symbolic expressions.
A typical algorithm begins with an initial estimate
\[ x_0 \]and generates
\[ x_1,\quad x_2,\quad x_3,\quad\ldots \]with the goal that the sequence approaches a desired solution.
The central questions become:
- Does the method approach the correct value?
- How quickly does it improve?
- How much error remains?
- When should the algorithm stop?
Residuals
When solving
\[ f(x)=0, \]a proposed approximation \(a\) can be checked by evaluating
\[ f(a). \]The quantity \(f(a)\) is called a residual.
If \(a\) is an exact root, then
\[ f(a)=0. \]For an approximate root, we often hope that
\[ |f(a)| \]is small.
Example: Checking an Approximate Root
Consider
\[ f(x)=x^2-2 \]and the approximation
\[ a=1.414. \]Evaluate:
\[ f(1.414) = 1.414^2-2. \] \[ = 1.999396-2. \] \[ \boxed{f(1.414)=-0.000604}. \]The residual is small, supporting the claim that \(1.414\) is close to a root.
A Small Residual Is Useful but Not the Whole Story
A small residual often indicates a good approximation, but residual size and solution error are not identical concepts.
The relationship between
\[ |f(a)| \]and
\[ |a-x| \]depends on the behavior of the function near the root.
Later numerical analysis develops more precise error estimates.
Verifying an Exact Solution
Exact solutions are typically verified by substitution or algebraic equivalence.
Suppose
\[ x=\sqrt7 \]is proposed as a solution of
\[ x^2=7. \]Substitute:
\[ (\sqrt7)^2=7. \]The equality holds exactly.
Verifying an Approximate Solution
Suppose
\[ x\approx2.646 \]is proposed as an approximation to the positive solution of
\[ x^2=7. \]Substitute:
\[ 2.646^2=7.001316. \]The result is close to \(7\), but not equal to it.
That is exactly what we expect from an approximate solution.
Exact Solutions and Approximate Data
It is important to distinguish an exact mathematical solution from the accuracy of the data placed into a model.
A formula may be manipulated exactly even when its inputs came from measurements.
For example, if a measured radius is entered as \(r=4.2\), then
\[ A=\pi(4.2)^2=17.64\pi \]is an exact expression relative to the entered value \(4.2\).
But if \(4.2\) itself was rounded from a physical measurement, the modeled area still inherits that measurement uncertainty.
Exact Arithmetic and Floating-Point Arithmetic
Computers often distinguish between exact symbolic arithmetic and finite numerical arithmetic.
Exact arithmetic may preserve
\[ \frac13 \]as a fraction.
Floating-point arithmetic instead stores a finite binary approximation of many real numbers.
This difference becomes important when numerical algorithms perform many operations.
Roundoff Error
Roundoff error arises because a computational system cannot represent every real number with unlimited precision.
A machine may store a nearby representable number instead.
A single roundoff error is often extremely small, but repeated calculations can sometimes amplify its effect.
Approximation Is Controlled, Not Careless
Numerical mathematics does not treat approximate answers as vague guesses.
A strong numerical solution specifies or controls:
- how the approximation was produced;
- what precision was requested;
- what stopping condition was used;
- what error or residual remains;
- how the result was verified.
Approximation becomes mathematically reliable when its uncertainty is understood and managed.
Choosing Exact or Approximate Form
Use an exact form when:
- symbolic structure matters;
- later algebraic manipulation is required;
- rounding would introduce unnecessary error;
- an exact expression is concise and meaningful.
Use an approximate form when:
- a decimal value is needed for interpretation;
- measurement or data already limit precision;
- no convenient exact form is available;
- a numerical algorithm is being used;
- a specified tolerance is sufficient for the application.
Example: Choosing the Better Form
Suppose a square has area
\[ A=10. \]Its side length is
\[ \boxed{s=\sqrt{10}}. \]This exact form is ideal for algebra.
If the side must be cut to the nearest millimeter after lengths have been expressed in centimeters, a decimal such as
\[ \boxed{s\approx3.16} \]may be more operationally useful.
Common Error: Replacing \(=\) with an Approximation Without Notice
Suppose
\[ x=\sqrt3. \]Writing
\[ x=1.732 \]incorrectly claims exact equality.
Write
\[ \boxed{x\approx1.732} \]when digits have been rounded.
Common Error: Assuming Every Decimal Is Approximate
The value
\[ 0.125 \]is exactly equal to
\[ \frac18. \]Therefore,
\[ \boxed{0.125=\frac18}. \]Decimal notation itself does not determine whether a value is exact.
Common Error: Reporting Too Many Digits
A calculator may display many digits, but the problem may not justify reporting all of them.
If measured data are given only to three significant figures, reporting a final answer with ten apparently precise digits can imply more certainty than the inputs support.
Precision should match the mathematical or practical context.
Common Error: Rounding Every Intermediate Step
Repeated early rounding can accumulate error.
Unless a procedure specifically requires otherwise, retain additional internal digits and round the final answer only once.
Common Error: Treating a Small Residual as Proof of Exactness
If
\[ f(a)=0.000001, \]then \(a\) is not an exact root unless the actual value of \(f(a)\) is exactly zero.
A small residual supports an approximation claim, not an exact equality claim.
A Reliable Workflow
- Identify whether the problem admits a useful exact solution.
- Preserve exact fractions, radicals, and constants when practical.
- If a decimal is needed, choose the required precision.
- Compute using sufficient internal precision.
- Round or truncate only according to the stated requirement.
- Use \(\approx\) when the reported value is approximate.
- Substitute or evaluate a residual to verify the result.
- State an error bound or tolerance when the application requires one.
Practice 1: Exact or Approximate?
Classify each representation as exact or approximate:
\[ \frac{5}{8}, \qquad 0.625, \qquad \sqrt3, \qquad 1.732. \]Practice 2: Use the Correct Symbol
Fill in \(=\) or \(\approx\):
\[ \pi\ \square\ 3.14159. \] \[ \frac34\ \square\ 0.75. \]Practice 3: Exact and Approximate Solution
Solve
\[ x^2=11 \]exactly, then approximate the solutions to three decimal places.
Practice 4: Rounding
Round
\[ 8.376492 \]to:
- two decimal places;
- four decimal places.
Practice 5: Truncation
Truncate
\[ 5.918734 \]to three decimal places.
Practice 6: Absolute Error
Use
\[ \pi\approx3.14. \]Write an expression for the absolute error and estimate it numerically.
Practice 7: Bracketing
Show that the positive solution of
\[ x^2=5 \]lies between \(2.2\) and \(2.3\).
Practice 8: Residual
For
\[ f(x)=x^2-5, \]compute the residual at
\[ x=2.236. \]Practice 9: Avoid Early Rounding
Explain why
\[ \sqrt5\cdot\sqrt5 \]should preferably be simplified exactly before replacing \(\sqrt5\) with a rounded decimal.
Practice 10: Interpretation
A numerical algorithm returns
\[ x\approx0.739085 \]for a solution of
\[ x=\cos x. \]Explain what the approximation symbol communicates.
Practice Answers
Practice 1
The fraction
\[ \frac58 \]is exact.
Because
\[ \frac58=0.625, \]the terminating decimal \(0.625\) is also exact.
The radical \(\sqrt3\) is exact.
The finite decimal \(1.732\) is an approximation to \(\sqrt3\).
Practice 2
\[ \boxed{\pi\approx3.14159}. \] \[ \boxed{\frac34=0.75}. \]Practice 3
The exact solutions are
\[ \boxed{x=\pm\sqrt{11}}. \]To three decimal places,
\[ \boxed{x\approx\pm3.317}. \]Practice 4
To two decimal places,
\[ \boxed{8.38}. \]To four decimal places,
\[ \boxed{8.3765}. \]Practice 5
Truncating after three decimal places gives
\[ \boxed{5.918}. \]Practice 6
The absolute error is
\[ |\pi-3.14|. \]Numerically,
\[ \boxed{|\pi-3.14|\approx0.00159265}. \]Practice 7
\[ 2.2^2=4.84<5 \]and
\[ 2.3^2=5.29>5. \]Therefore,
\[ \boxed{2.2<\sqrt5<2.3}. \]Practice 8
\[ f(2.236) = 2.236^2-5. \] \[ = 4.999696-5. \] \[ \boxed{f(2.236)=-0.000304}. \]Practice 9
Exact simplification gives
\[ \boxed{\sqrt5\cdot\sqrt5=5}. \]Replacing \(\sqrt5\) by a rounded decimal first introduces unnecessary approximation error into a calculation whose exact value is already known.
Practice 10
The statement
\[ x\approx0.739085 \]means that the displayed decimal is close to the true solution but is not being asserted to contain the complete exact value.
What to Remember
- An exact value preserves the full mathematical value.
- An approximate value replaces the exact value with a nearby number.
- Terminating and explicitly repeating decimals can be exact.
- Use \(=\) for equality and \(\approx\) for approximation.
- Keep exact form as long as practical when symbolic work remains.
- Round according to the precision required by the problem.
- Absolute error measures the distance between an exact value and an approximation.
- A tolerance specifies how much error is acceptable.
- Residuals help check approximate solutions of equations.
- Numerical methods make approximation systematic, measurable, and verifiable.
Where We Go Next
Exact and approximate solutions establish the basic language of numerical mathematics. The next step is to organize numerical information so that a solution can be located and refined systematically.
The next lesson, Numerical Tables and Successive Approximation, develops table-based methods for evaluating functions, narrowing search regions, and producing progressively better estimates.