C1.13; E1.13 - Using a graphic display calculator

C1.13; E1.13 - Using a graphic display calculator

Why can two almost identical entries, (−4)2(-4)^2 and −42-4^2, produce opposite signs? A calculator follows the expression you enter, including its grouping. This lesson develops accurate entry, retained working precision and interpretation of money and time displays in C1.13 and E1.13.

Entering calculations

A fraction bar groups its whole numerator and denominator. In a linear entry, replace that grouping with brackets. A calculator's fraction template can do the same job, provided that you leave each box before starting the next operation.

Use the signed-number key when entering a negative value; it may be different from the subtraction key. Check the displayed expression before pressing equals. The square of −4-4 needs brackets: (−4)2=16(-4)^2=16, but −42=−(42)=−16-4^2=-(4^2)=-16.

You already know

Brackets control grouping, then powers are evaluated before multiplication, division, addition and subtraction. A calculator implements these rules; it does not infer the expression you intended.

Different models use different key sequences. The expression on screen is the reliable check, rather than memorising a sequence from another calculator.

Worked example 1: squaring a negative value

Question. Calculate (−4)2+65\dfrac{(-4)^2+6}{5}.

Plan. Group the negative base, then group the complete numerator before dividing.

Enter ((−4)2+6)÷5\bigl((-4)^2+6\bigr)\div5, or use a fraction template with the complete numerator above the bar.

(−4)2=16the negative value is squared16+65=225=4.4\begin{aligned} (-4)^2&=16 &&\text{the negative value is squared}\\ \frac{16+6}{5}&=\frac{22}{5}=4.4 \end{aligned}

Answer. 4.44.4.

Check. 4.4×5=224.4\times5=22, and 22−6=1622-6=16, the square of −4-4.

Remove only the brackets around the negative base. The minus sign now sits outside the square, which changes the numerator.

Worked example 2: a minus sign outside a power

Question. Calculate −42+65\dfrac{-4^2+6}{5}.

Plan. Keep the complete numerator grouped, but square 4 before applying its preceding minus sign.

Enter (−(42)+6)÷5\bigl(-(4^2)+6\bigr)\div5. The extra grouping makes the intended order explicit even on a model with an unfamiliar signed-entry convention.

−42=−16power before the preceding minus−16+65=−105=−2\begin{aligned} -4^2&=-16 &&\text{power before the preceding minus}\\ \frac{-16+6}{5}&=\frac{-10}{5}=-2 \end{aligned}

Answer. −2-2.

Check. (−2)×5−6=−16(-2)\times5-6=-16, matching −42-4^2. The negative numerator divided by a positive denominator should give a negative answer.

These calculations are also small enough to perform by hand, which provides a useful independent check. A calculator answer still needs the written expression and relevant working.

For this course, a graphic display calculator is required on Papers 3–6 and calculators are prohibited on Papers 1 and 2. Later lessons teach its permitted graph, table, point-plotting, zero, extremum, intersection and statistical functions; linear regression is Extended only. Other built-in applications, such as an equation-solver application, gain no credit. Calculators with symbolic algebraic logic and applications or programs from external sources are not permitted.

Watch out

Entering (−4)2+6÷5(-4)^2+6\div5 leaves the division attached only to 6. For the fraction in the first example, both terms belong above the bar: enter ((−4)2+6)÷5\bigl((-4)^2+6\bigr)\div5.

Inspect the complete numerator and denominator before evaluating.

Keeping precision

The displayed digits are a view of a calculator's value, rather than an instruction to copy and shorten it. Use the previous-answer function or a stored value to carry an intermediate result forward. Entering the whole calculation in one expression is often simpler.

Working precision

Working precision is the accuracy kept during a calculation. Preserve the calculator's unrounded intermediate values, then round the final answer to the requested accuracy.

Write enough digits in your working to explain the result, while continuing with the stored value rather than re-entering the written approximation.

Worked example 3: reusing a quotient

Question. Calculate (14÷9)×7(14\div9)\times7, giving your answer correct to three significant figures.

Plan. Keep the full quotient in memory, multiply by 7, then round.

14÷9=1.555555…(14÷9)×7=10.888888…≈10.9three significant figures\begin{aligned} 14\div9&=1.555555\ldots\\ (14\div9)\times7&=10.888888\ldots\\ &\approx10.9 &&\text{three significant figures} \end{aligned}

On a calculator, enter the whole expression or multiply the previous answer by 7. Rounding the quotient to 1.56 first would instead give 10.92.

Answer. 10.910.9.

Check. Exactly, (14/9)×7=98/9=10+8/9(14/9)\times7=98/9=10+8/9, which agrees with the unrounded display.

Now square the quotient before using it. A small early error is used twice in the multiplication, so preserving the original value matters even more.

Worked example 4: a stored value used in a power

Question. Calculate (14÷9)2×81(14\div9)^2\times81.

Plan. Enter the whole expression, or square the stored quotient before multiplying by 81.

(14÷9)2×81=2.419753086…×81=196\begin{aligned} (14\div9)^2\times81 &=2.419753086\ldots\times81\\ &=196 \end{aligned}

Using 1.56 in place of 14/914/9 would give 1.562×81=197.12161.56^2\times81=197.1216, a different answer.

Answer. 196196, exactly.

Check. By hand, (149)2×81=19681×81=196\left(\frac{14}{9}\right)^2\times81=\frac{196}{81}\times81=196. This confirms that the exact whole number is not a rounded display accident.

If no different accuracy is specified, calculator papers normally require non-exact answers to three significant figures, or degree angles to one decimal place. An exact result such as 12.345 is not automatically rounded to three significant figures. Keep exact π\pi or surd forms when requested in later topics, rather than replacing them with decimal approximations.

Watch out

Do not re-enter a rounded intermediate answer. Use the stored unrounded value even when an earlier part asked you to write that value to a stated accuracy.

Carry the value through the entire expression.

Interpreting displays

A calculator shows numbers without knowing their context. A display of 4.8 dollars means $4.80. A display of 3.25 hours means 3 hours and a quarter of an hour, which is 15 minutes, rather than 25 minutes.

Decimal hours

In decimal hours, the decimal part is a fraction of one hour. Multiply that part by 60 to obtain minutes. To convert hours and minutes into decimal hours, divide the minutes by 60 and add the whole hours.

For example, 2 hours 30 minutes is 2+30/60=2.52+30/60=2.5 hours. Some calculators offer a sexagesimal input written with degree, minute and second symbols. Entering 2∘30′0′′2^\circ30'0'' in that numerical conversion facility represents 2+30/60+0/3600=2.52+30/60+0/3600=2.5. For a time calculation the result's unit is hours, not an angle: check the calculator's conversion instructions and do not type 2.30 as a substitute.

Worked example 5: interpreting a charge

Question. Parking costs $2.40 per hour. A car is parked for 2 hours 15 minutes. Calculate the charge.

Plan. Convert the duration to decimal hours, multiply, then write the money display in cents.

time=2+1560=2.25 hcharge=2.40×2.25=5.4\begin{aligned} \text{time}&=2+\frac{15}{60}=2.25\,\mathrm{h}\\ \text{charge}&=2.40\times2.25=5.4 \end{aligned}

Answer. $5.40.

Check. Two hours cost 4.80andaquarter−hourcosts4.80 and a quarter-hour costs 0.60, totalling $5.40. The display 5.4 has not lost ten cents.

Now read a decimal duration in the opposite direction. Money uses hundredths of a dollar; time uses sixtieths of an hour, so the decimal digits need interpreting differently.

Worked example 6: interpreting a duration

Question. A calculation gives a journey time of 1.35 hours. Write the time in hours and minutes.

Plan. Separate the whole hour and convert the fractional hour to minutes.

1.35 h=1 h+0.35 h0.35×60=21 min\begin{aligned} 1.35\,\mathrm{h}&=1\,\mathrm{h}+0.35\,\mathrm{h}\\ 0.35\times60&=21\,\mathrm{min} \end{aligned}

Answer. 1 hour 21 minutes.

Check. 1+21/60=1.351+21/60=1.35 hours. Writing 1 hour 35 minutes would convert back to 1.58333…1.58333\ldots hours, so it cannot match the display.

The opening sign puzzle is now resolved by the same habit used for time and money: inspect what the entry or display means before acting on it.

Watch out

2 hours 30 minutes is 2.5 hours, not 2.30 hours. In decimal hours, 2.30 means 2 hours 18 minutes because 0.30×60=180.30\times60=18.

Use sixtieths for the time conversion.

Quick Recap

  • Enter the expression's grouping, including complete numerators, denominators and signed bases.
  • Keep intermediate values stored at full precision and round only at the end.
  • A money display of 4.8 means $4.80.
  • Decimal hours use fractions of 60 minutes; 2 hours 30 minutes is 2.5 hours.

In the opening pair, (−4)2=16(-4)^2=16 but −42=−16-4^2=-16: brackets decide whether the minus sign is squared.