Rounding, Pi, and Calculator Techniques (TI-83/TI-84)

Rounding and Decimal Places

• Rounding is used to simplify numbers so they fit the required precision (e.g., two, four, or zero decimal places).

• Pi (π) is an irrational number: its decimal expansion goes on forever without repeating. A quick analogy used: if your roommate were irrational, you wouldn’t know what they’d do next; similarly, the digits of π go on forever and don’t settle into a repeating pattern.

• When numbers fill the screen or data needs to be reported with limited precision, rounding is essential.

• Common rounding scenarios shown:

  • Round to four decimal places: 3.1416
  • Round to three decimal places: 3.142
  • Round to two decimal places: 3.14
  • Round to one decimal place: 3.1
  • Round to zero decimal places (nearest whole number): 3

• Rounding rule (the last digit check):

  • Find the target decimal place (k).
  • Look at the next digit (the (k+1)th decimal place).
  • If that next digit is 5 or bigger, bump the last kept digit by 1; otherwise, leave it.
  • Always only worry about the last kept digit; all earlier digits stay the same.
  • The process is the same whether rounding to 2, 3, 4 decimals, or to zero decimals.

• “Approximate” notation:

  • When a value is not exact, we write it as approximately equal: $\approx$ (tilde symbol).
  • Example: $\pi \approx 3.1416$ (when rounded to four decimals).

• Rounding in real examples:

  • Rounding to four decimals: $\pi \approx 3.1416$
  • Rounding to three decimals: $\pi \approx 3.142$
  • Rounding to two decimals: $\pi \approx 3.14$
  • Rounding to one decimal: $\pi \approx 3.1$
  • Rounding to zero decimals: $\pi \approx 3$

• Rounding in context:

  • In statistics and many courses, you often round to four decimals for exactness, but in everyday calculations you might round to two decimals.
  • For problems involving animals (e.g., deer on a preserve), you round to the nearest whole number because you cannot have a fraction of an animal.
  • Money often uses two decimal places for cents.

• How to round step-by-step (recap):

  1. Identify the decimal place you are rounding to.
  2. Underline or focus on that many digits to keep.
  3. Check the next digit; if it is 5 or bigger, increase the last kept digit by 1; otherwise, keep it as is.
  4. Drop all digits beyond the kept place.

• The teacher uses a playful, practical approach to rounding, emphasizing practice to reduce errors across many problems throughout the course.

Pi and Euler’s Number (e)

• π (pi):

  • Defined as the ratio of a circle’s circumference to its diameter.
  • Irrational and transcendental; its decimal expansion is infinite and non-repeating.
  • Common approximations: $\pi \approx 3.1416\, (to~4~decimals)$, $3.142\, (to~3~decimals)$, etc.

• e (Euler’s number):

  • The base of natural logarithms; arises in continuous growth/decay and compound interest as the number for which growth is most 'natural' in continuous time.
  • Approximate value (not memorized here): \mathrm{e} \approx 2.7182818… \approx 2.72\n
  • Location on a TI calculator (convenience note): on many models, e is accessed by pressing the 2nd function of the division key (often labelled ÷) to yield the constant $\mathrm{e}$. The speaker emphasizes not memorizing the long digits, just knowing where to find it.
  • Historical note: Euler studied compound interest among other problems and defined e as a limit related to growth rates over time.

• Practical takeaway:

  • You will use π when dealing with circles and geometry.
  • You will use e in problems involving exponential growth/decay and continuous compounding ( Chapter 4 context in the course).

Calculator Tips: TI-83 vs TI-84, and Basic Operations

• Locating and using e:

  • TI-83/84: e is often found by pressing 2nd and then the division symbol key; the result is the constant $\mathrm{e} \approx 2.71828…$
  • You do not need to memorize all digits; just know where to retrieve e.

• Clearing and backing out:

  • If you press something you didn’t intend, you can press 2nd then QUIT to back out and reset the screen; or use CLEAR to clear the screen; turning the calculator off and back on also resets it.

• Approximate display:

  • The calculator may show exact values or approximations; for many problems you round after computing.

• The keyboard interface differences mentioned:

  • TI-83 (the speaker calls it the “dumb calculator” for input-pun):
  • It requires explicit opening and closing of parentheses (opening a door, then closing it) to enforce order of operations.
  • Fractions on the 83 are more cumbersome; fractions and nested operations require careful use of parentheses.
  • TI-84 (the upgraded model):
  • Has a dedicated fraction input button/template, often accessed via a green key like alpha followed by F1 (the exact key may vary by model).
  • The fraction template allows you to input the numerator and denominator cleanly; press Enter to evaluate.
  • It’s also noted that there are additional functions (such as a feature that can draw a picture; teacher uses alpha F1 for that third function). For standard work, use the fraction template for clean fractions and the standard operators for other functions.

• Quick power and square roots:

  • For squaring, you can use the x^2 button or the general exponent function (the boomerang arrow on some models): e.g., to compute $3^2 + 4^3$, you can:
  • Long way: enter 3, press the ^2 button to square; then +; enter 4, press the ^3 button to cube; press Enter.
  • Short way: use dedicated square (x^2) and cube (x^3) keys where available; ensure you step out of the power so you don’t keep applying the exponent to the whole expression.
  • Important: when using exponents, you must exit the exponent context (step out) before continuing with the rest of the expression.

• Notes on roots:

  • The square root button opens a root; you must close the root (i.e., close the parentheses) to stop applying the root to additional terms.
  • Example: inputting $\sqrt{25} + 4$ requires closing the root after 25 before adding 4; otherwise you may accidentally take the square root of more than intended.

• Fractions and order of operations on the TI-83 vs TI-84:

  • 83: Fractions are input with a fraction-like approach using parentheses to ensure proper grouping. If you do 9 + 5 ÷ 3 + 4 without parentheses, the order of operations may produce an incorrect result.
  • 84: Use the dedicated fraction button (alpha F1) to input a proper fraction template (numerator over denominator). It simplifies input and reduces error.

• A few practical tips from the session:

  • Always check the next digit when rounding; that last digit will determine rounding up or staying the same.
  • Keep laptop practices clean in class (close laptops when not needed) to focus on the calculator work.
  • Practice with simple problems first (e.g., sqrt(25) + 4, 3^2 + 4^3) to get comfortable with input habits.

Fractions, Order of Operations, and Input Nuances

• Order of operations governs the evaluation order: perform inside parentheses first, then exponents, then multiplication/division (from left to right), then addition/subtraction (from left to right).

• For a fraction like

  • On the 83: you must explicitly structure as (numerator) ÷ (denominator) with parentheses to enforce the intended grouping.
  • On the 84: you can use the fraction template to enter this as a clean horizontal fraction, which enforces the same grouping visually and in calculation.

• Example provided in the session for a fraction:

  • Using a 83-style approach: evaluate (9 + 5) ÷ (3 + 4) to obtain 2.
  • Using a 84-style approach: use the fraction template to input the same structure cleanly and avoid misgrouping.

• The professor’s highlighted tips for fractions:

  • Always use proper doors (parentheses) when inputting fractions with multiple operations inside the numerator or denominator.
  • On 84, the fraction template (alpha F1) makes this easier and reduces input errors.

Function Notation, Initial Value, and “Crystal Ball” Problems

• Function notation basics:

  • f(x) denotes a function with input x and output depending on x.
  • h(x, y, z) denotes a function with three inputs; you plug in x, y, z into their respective slots.
  • Function notation is a narrative device in problems: e.g., f(x) could represent “the number of fish in a pond after x weeks.”

• Interpreting a story with a crystal ball: using a function to predict quantities over time

  • Example narrative: the function tells you how many fish you have after x weeks.
  • To compute for a specific week, substitute that week value for x and evaluate the expression.
  • If the problem has more variables (x, y, z), substitute each input into its corresponding slot (x for the first, y for the second, z for the third).

• Initial value (initial condition) concept:

  • Initial value means the quantity at time zero (t = 0) or the starting amount before any time has passed.
  • In the fox population example, the initial value is how many foxes you start with at time zero (e.g., 87,300,000). This is a common phrase in problems: initial value, initial concentration, starting amount, etc.
  • When a problem asks for the initial value, plug in 0 for time to obtain the starting quantity.

• Exponential growth/decay form (example structure):

  • A typical continuous-growth model uses the form: $N(t) = N_0 \, e^{k t}$ where:
  • $N(t)$ is the quantity after time $t$,
  • $N_0$ is the initial value (initial quantity at $t=0$),
  • $k$ is the growth (positive) or decay (negative) rate,
  • and $e$ is Euler’s number.
  • In the transcript, a scenario is discussed with a base value and an exponential term like $e^{5t}$, where the exponent is a product of a coefficient and time (e.g., $5t$ means $5 \times t$).
  • Multiplication between a number and a variable (e.g., $5t$) is implied when a number and a variable are adjacent; this is important for input into a calculator or algebraic manipulation.

• Practical tips for interpreting multi-variable function problems:

  • When given a function h(x, y, z) with three inputs, plug the provided numbers into the corresponding slots in the order listed.
  • If a problem describes a scenario (e.g., weight of a horse given inputs for corn, oats, and hay), treat the expression as a story and map each input to its variable (x, y, z) in the function.

• The narrative approach is a teaching tool:

  • By imagining a story (fish population, deer counts, cattle herds, horses, etc.), you can more easily remember what the inputs represent and how to substitute values.

Practice Problems: Crystal Ball Scenarios and Two-Decimal Rounding

• The crystal ball idea: every problem is a function that predicts a future quantity as time passes. You replace the time variable with the given value and compute.

  • Example narrative: f(x) represents the number of fish after x weeks; plug in x = 1 for one week later, x = 2 for two weeks later, etc., then use a calculator to evaluate.
  • You should then round the final result to the required precision (often two decimal places for reports).

• Worked example structure (as described in class):

  • Step 1: Write the function and identify the time variable (e.g., f(x) for fish after x weeks).
  • Step 2: Substitute the requested time value (e.g., x = 1) into the expression.
  • Step 3: Use the calculator to evaluate the expression.
  • Step 4: Round the result to the requested precision, checking the next digit to decide if rounding up is warranted.
  • Step 5: State the final rounded answer, e.g., “The number of fish after one year is about $1.32\times 10^3$ (rounded to two decimals).”

• An example provided in the lecture (narrative form):

  • A fox population model is given with exponential terms (e.g., something like $N(t) = N_0 + \ldots e^{5t} \ldots$) where the initial value is stated (e.g., 87,300,000 foxes) and the population changes over time.
  • The class discusses inputting the coefficients and exponents carefully, emphasizing multiplication like $5t$ and interpreting the initial value as the quantity at time zero.

• Final reminder on multi-input functions (e.g., h(x, y, z)):

  • Treat the three inputs as three separate numbers to be placed into the corresponding slots of the function notation.
  • For three-input problems, substitute the first number into x, the second into y, and the third into z. The process is the same as for single-input functions; just keep track of which input goes where.

• Quick recap on the main takeaways from the transcript:

  • Rounding is a pervasive skill throughout the course; always check the next digit before finalizing the rounded value.
  • π is irrational; e is the base of natural logarithms; both play foundational roles in different areas of mathematics.
  • The TI-83 requires careful input with explicit parentheses and careful handling of roots; the TI-84 offers easier fraction input and some additional features.
  • Function notation is a storytelling tool; initial value and multi-variable inputs follow the same substitution principle with careful attention to which input corresponds to which variable.