Scientific Notation on TI-84 and Casio fx-991: A Precision Reference

Scientific notation is the universal language for expressing very large or very small numbers in science and engineering. The TI-84 and Casio fx-991 are two of the most widely used calculators in education and professional practice, yet their handling of scientific notation—including display formats, rounding, and significant figures—often leads to confusion. This reference consolidates the rules, conventions, and standards governing scientific notation on these devices, serving as a definitive resource for students, engineers, and metrologists.

Rule Statement

Scientific notation expresses a number as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. The coefficient a carries all significant digits, and the exponent n indicates the order of magnitude. When rounding a value to k significant figures in scientific notation, apply the standard rounding rules to the coefficient a only; the exponent remains unchanged unless the rounding causes a carry (e.g., 9.99 × 10³ rounded to 2 sig figs becomes 1.0 × 10⁴).

Both the TI-84 and Casio fx-991 follow the IEEE 754 double-precision floating-point standard internally, but they differ in how they display and accept scientific notation. The TI-84 uses E-notation (e.g., 1.23E-4), while the Casio uses the traditional ×10ⁿ display. Input conventions also differ: the TI-84 uses the EE key (2nd + comma), while the Casio uses the ×10x key.

For rounding, both calculators default to round-half-up (also called symmetric rounding) when displaying a result to a specified number of digits. However, the underlying binary representation may introduce subtle differences in edge cases, as discussed in the Software Behavior Note.

Worked Examples

Example 1: Converting a Decimal to Scientific Notation

Problem: Express 0.00012345 in scientific notation with 3 significant figures on both calculators.

1. Identify the coefficient: Move the decimal point so that the first non-zero digit is to the left. For 0.00012345, the coefficient is 1.2345, and the exponent is -4 (since we moved the decimal 4 places to the right).
2. Round to 3 significant figures: The coefficient 1.2345 rounded to 3 sig figs is 1.23 (the fourth digit is 4, so no rounding up).
3. Result: 1.23 × 10^-4.

On the TI-84: Press 2nd EE to enter the exponent. Type 1.23 then 2nd EE then -4. The display shows 1.23E-4.

On the Casio fx-991: Type 1.23, press ×10x, then type -4. The display shows 1.23×10⁻⁴.

Example 2: Rounding in Scientific Notation with Carry

Problem: Round 9.97 × 10⁵ to 2 significant figures.

1. Round the coefficient: 9.97 rounded to 2 sig figs is 10 (since the third digit is 7 ≥ 5, we round up).
2. Normalize: 10 × 10⁵ is not in proper scientific notation (coefficient must be < 10). Convert to 1.0 × 10⁶.
3. Result: 1.0 × 10⁶ (with the zero significant).

Both calculators handle this automatically when using the SCI display mode; they will show 1.0E6 or 1.0×10⁶ after rounding.

Example 3: Multiplication and Division

Problem: Compute (2.5 × 10³) × (4.0 × 10^-2) and report the answer with the correct number of significant figures.

Rule: In multiplication/division, the result has the same number of significant figures as the factor with the fewest sig figs. Here, both factors have 2 sig figs, so the result must have 2 sig figs.

1. Multiply coefficients: 2.5 × 4.0 = 10.0
2. Add exponents: 10³ × 10^-2 = 10¹
3. Preliminary result: 10.0 × 10¹ = 1.00 × 10² (normalized).
4. Round to 2 sig figs: 1.0 × 10² (the third digit 0 is not significant, but we keep the zero as a placeholder).

On the TI-84, entering 2.5E3 * 4.0E-2 yields 100 in normal mode, but switching to SCI mode displays 1.0E2 if the display is set to 2 significant digits. On the Casio, the same result appears as 1.0×10².

Counter-Examples

Counter-Example 1: Misinterpreting E-Notation as a Variable

Error: On the TI-84, typing 1.23E-4 without using the EE key treats E as a variable (if defined) or as an error. The correct input is 1.23 then 2nd EE then -4.

Consequence: The calculator may evaluate 1.23 * E - 4 if E is stored, leading to a completely different numeric result.

Counter-Example 2: Double Rounding

Error: Rounding a value to 3 sig figs by first rounding to 4 sig figs and then to 3. For example, 1.2345 × 10² rounded to 3 sig figs should be 1.23 × 10² (since the fourth digit is 4). But if you first round to 4 sig figs (1.235) and then to 3 (1.24), you get an incorrect result.

Correct approach: Round directly from the original value. Calculators do this internally, but users may inadvertently double-round when using intermediate display steps.

Counter-Example 3: Ignoring Significant Zeros

Error: Reporting 1.0 × 10³ as 1 × 10³ loses the significant zero. The zero in 1.0 is significant and indicates a precision of two sig figs. Omitting it changes the meaning.

Calculator behavior: In SCI mode, the TI-84 and Casio will display the trailing zero if the display precision is set to 2 significant digits, but they may drop it if set to 1 digit. Always check the display setting.

Convention Comparison Table

Standards Citation

Adherence to recognized standards ensures consistency in scientific communication. The following standards are directly relevant to scientific notation and rounding:

• ASTM E29-13 – Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. Section 6.2 defines the rounding method for numbers expressed in scientific notation, specifying that the coefficient is rounded to the desired number of significant digits using the “round half up” rule (unless otherwise specified).
• ISO 80000-1:2022 – Quantities and units – Part 1: General. Clause 7.3.2 recommends the use of scientific notation with a mantissa between 1 and 10, and the multiplication sign (×) or a dot (·) for the product. It also specifies that the exponent should be an integer.
• JCGM 100:2008 (GUM) – Evaluation of measurement data – Guide to the expression of uncertainty in measurement. Clause 7.2.6 advises that when reporting uncertainty, the numerical value of the uncertainty should be given to two significant figures, and the result should be expressed in scientific notation when the magnitude is very large or very small.
• NIST SP 811 – Guide for the Use of the International System of Units (SI). Section 7.1 discusses the use of scientific notation and the proper formatting of powers of ten.

When using calculators, always verify that the displayed result conforms to the required standard. For example, ASTM E29 requires that the rounding be performed on the coefficient without intermediate rounding of the exponent.

Common Mistakes

• Forgetting to normalize: After rounding, the coefficient may become ≥ 10 (e.g., 9.99 → 10). Always re-normalize to 1 ≤ |a| < 10.
• Misreading E-notation: On the TI-84, 1.23E-4 is often misread as 1.23 times E minus 4. Remember that E stands for “times ten to the power of.”
• Setting the wrong number of significant digits: The display precision on both calculators is set independently of the internal precision. If you set 3 digits, the calculator will round the coefficient to 3 significant digits, but the exponent may change if a carry occurs.
• Using engineering notation when scientific is required: Engineering notation uses exponents that are multiples of 3, which may not align with significant figure rules. Always check the mode.
• Assuming exact representation: Both calculators use binary floating-point, so some decimal fractions (e.g., 0.1) are not exactly representable. This can cause tiny errors that affect rounding in edge cases.
• Ignoring the role of trailing zeros: In scientific notation, trailing zeros in the coefficient are significant. Ensure the display includes them when required.

Software Behavior Note

The TI-84 and Casio fx-991 both use a binary floating-point format internally, but their display logic differs. The TI-84 stores numbers in a proprietary format that often exhibits round-half-up behavior when converting to decimal for display. The Casio fx-991 uses a similar approach but may employ a slightly different rounding algorithm for its 24-digit internal precision (which is then rounded to the display precision).

In practice, both calculators will round to the nearest decimal value at the specified number of significant digits. However, because binary fractions cannot exactly represent many decimal numbers, a value like 1.005 may be stored as 1.004999999… and thus round down to 1.00 when displayed to 3 sig figs, whereas the expected half-up rounding would give 1.01. This is a known limitation of all binary calculators. Users should be aware that the calculator’s displayed result is the final authority for that device, but for critical work, manual rounding from the full precision value is recommended.

Both calculators also support engineering notation (ENG mode), which displays exponents as multiples of 3 (e.g., 1.23×10³, 4.56×10⁶). This is useful for SI prefixes but is not a substitute for scientific notation when significant figures are concerned.

Quick Reference Table

Sources & Further Reading

• ASTM E29-13, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, ASTM International.
• ISO 80000-1:2022, Quantities and units – Part 1: General, ISO.
• JCGM 100:2008, Evaluation of measurement data – Guide to the expression of uncertainty in measurement (GUM), BIPM.
• NIST SP 811, Guide for the Use of the International System of Units (SI), NIST.
• TI-84 Plus CE Graphing Calculator Guidebook, Texas Instruments.
• Casio fx-991EX User’s Guide, Casio.

For more detailed guidance on rounding and significant figures, see our Rounding Rules and Sig Figs in Scientific Notation articles.