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Classic Graphing 84 manual · Chapter 4 of 25

Functions: roots, logs, trig, probability

Powers and roots, logarithms, trigonometry, rounding and number tools, probability, comparisons and logic.

4.1Powers, squares and roots#

To getPressExample
A squarex² after the number5² = 25
Any power^ then the exponent2^10 = 1024
A reciprocalx⁻¹ after the number2⁻¹ = .5
A square root2nd√x² then the number and )√(16) = 4
A cubemath then 32 cubed = 8
A cube rootmath then 4³√(27) = 3
An x-th rootthe index, math, 5, then the number4 ˣ√ 81 = 3
a1/n=an,a−1=1aa^{1/n}=\sqrt[n]{a},\qquad a^{-1}=\frac1a
Roots and reciprocals as powers.
Example 4.1

Raise to a power

  1. Type 2 ^ 10.

    2^10enter
Result of the example "Raise to a power" The display shows: 2^10 / 1024.
You should see1024
Example 4.2

Take a reciprocal

  1. Type 2 then the x⁻¹ key.

    2x⁻¹enter
Result of the example "Take a reciprocal" The display shows: 2⁻¹ / .5.
You should see2⁻¹ .5
Example 4.3

Cube root

  1. Press math, 4, type 27 and close the bracket.

    math427)enter
Result of the example "Cube root" The display shows: ³√(27) / 3.
You should see³√(27) 3
Example 4.4

The fourth root of 81

  1. Type 4, press math then 5, and type 81.

    4math581enter
Result of the example "The fourth root of 81" The display shows: 4ˣ√81 / 3.
You should see3
Example 4.5

Cube a number

  1. Type 2, then math and 3 for the cube sign.

    2math3enter
Result of the example "Cube a number" The display shows: 2³ / 8.
You should see8

Fractional exponents also work: 9^.5 is 3. A negative number raised to a fractional power gives the real root when it exists (for example (−8)^(1/3) is −2).

4.2Logarithms and exponentials#

log is the base-10 logarithm and ln is the natural logarithm (base e). Their inverses are above them: 2nd10ˣlog is 10 to a power and 2ndeˣln is e to a power. For any other base use logBASE( in the MATH menu: base after the number.

log⁡bx=ln⁡xln⁡b\log_b x=\frac{\ln x}{\ln b}
Any logarithm from natural logs.
Example 4.6

Base-10 logarithm

  1. Type log(1000).

    log1000)enter
Result of the example "Base-10 logarithm" The display shows: log(1000) / 3.
You should seelog(1000) 3
Example 4.7

Natural logarithm of e

  1. Type ln(, then 2nd then the ÷ key for e, then close the bracket.

    ln2nde÷)enter
Result of the example "Natural logarithm of e" The display shows: ln(e) / 1.
You should see1
Example 4.8

A logarithm in any base

  1. Press math then alpha math (the letter A) to choose logBASE(. Type the number, a comma, the base and a bracket: log base 2 of 8.

    mathalphaAmath8,2)enter
Result of the example "A logarithm in any base" The display shows: logBASE(8,2) / 3.
You should seelogBASE(8,2) 3
Example 4.9

Ten to a power

  1. Press 2nd then log, type 2 and close the bracket.

    2nd10ˣlog2)enter
Result of the example "Ten to a power" The display shows: 10^(2) / 100.
You should see100

4.3Trigonometry and angle units#

Sine, cosine and tangent are on sin, cos and tan, and the inverse functions are on the same keys after 2nd. They use the angle unit chosen in MODE: Radian (the default) or Degree. If a trig answer looks wrong, check this setting first. Each key types the function name and an opening bracket; close the bracket yourself.

sin⁡θ=oppositehypotenuse,π rad=180∘\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \pi\text{ rad}=180^\circ
Radians and degrees.
Example 4.10

Sine in radians

  1. Type sin(π/6).

    sin2ndπ^÷6)enter
Result of the example "Sine in radians" The display shows: sin(π/6) / .5.
You should seesin(π/6) .5
Example 4.11

Sine in degrees

  1. Open MODE, move down to the angle row, move right to DEGREE and press enter, then return home.

    mode▼▼▶enter2ndQUITmode

  2. Type sin(30).

    sin30)enter
Result of the example "Sine in degrees" The display shows: sin(30) / .5.
You should seesin(30) .5
Example 4.12

Inverse sine

  1. Press 2nd then sin and type 1.

    2ndsin⁻¹sin1)enter
Result of the example "Inverse sine" The display shows: sin⁻¹(1) / 1.570796327.
You should see1.570796327
Example 4.13

Find an angle in degrees

  1. Switch to Degree mode.

    mode▼▼▶enter2ndQUITmode

  2. Type cos⁻¹(.5).

    2ndcos⁻¹cos.5)enter
Result of the example "Find an angle in degrees" The display shows: cos⁻¹(.5) / 60.
You should see60
Example 4.14

Mark one angle as degrees without changing the mode

  1. Press 2nd apps then 1 for the degree mark. Type sin(90°).

    sin902ndANGLEapps1)enter
Result of the example "Mark one angle as degrees without changing the mode" The display shows: sin(90°) / 1.
You should seesin(90°) 1

See also Find sine, cosine and tangentFind an angle from a ratioModes and settings: Radian or degree

4.4Absolute value, rounding and divisibility (NUM menu)#

Press math then ▶ for the NUM tab.

ItemWhat it givesExample
1:abs(Absolute valueabs(⁻7) = 7
2:round(Rounds to a number of decimalsround(π,3) = 3.142
3:iPart(Integer part (drops the decimals)iPart(⁻3.7) = ⁻3
4:fPart(Fraction partfPart(3.25) = .25
5:int(Greatest integer not above the number (floor)int(⁻3.7) = ⁻4
6:min(Smaller of two numbers (or of a list)min(4,9) = 4
7:max(Larger of two numbers (or of a list)max(4,9) = 9
8:lcm(Least common multiplelcm(4,6) = 12
9:gcd(Greatest common divisorgcd(12,18) = 6
0:remainder(Remainder after divisionremainder(17,5) = 2
The NUM tab of the MATH menu. The display shows: MATH [NUM] CMPLX PROB FRAC / 1:abs( / 2:round( / 3:iPart( / 4:fPart( / 5:int( / 6:min( / 7↓max(.
The NUM tab of the MATH menu.
Example 4.15

Absolute value

  1. Press math, right, 1 and type −7.

    math▶1(−)7)enter
Result of the example "Absolute value" The display shows: abs(⁻7) / 7.
You should seeabs(⁻7) 7
Example 4.16

Round to 3 decimals

  1. Press math, right, 2 and type π, 3.

    math▶22ndπ^,3)enter
Result of the example "Round to 3 decimals" The display shows: round(π,3) / 3.142.
You should see3.142
Example 4.17

The difference between iPart and int

  1. iPart(−3.7) drops the decimals.

    math▶3(−)3.7)enter

  2. int(−3.7) rounds down to the lower integer.

    math▶5(−)3.7)enter
Result of the example "The difference between iPart and int" The display shows: iPart(⁻3.7) / ⁻3 / int(⁻3.7) / ⁻4.
You should see⁻3 ⁻4
Example 4.18

Greatest common divisor

  1. Press math, right, 9 and type 12, 18.

    math▶912,18)enter
Result of the example "Greatest common divisor" The display shows: gcd(12,18) / 6.
You should seegcd(12,18) 6
Example 4.19

Remainder

  1. Press math, right, 0 and type 17, 5.

    math▶017,5)enter
Result of the example "Remainder" The display shows: remainder(17,5) / 2.
You should seeremainder(17,5) 2
Example 4.20

Least common multiple

  1. Press math, right, 8 and type 4, 6.

    math▶84,6)enter
Result of the example "Least common multiple" The display shows: lcm(4,6) / 12.
You should see12

4.5Probability: factorials, permutations, combinations, random numbers#

Press math then ◀ twice (or ▶ three times) for the PROB tab.

The PROB tab. The display shows: MATH NUM CMPLX [PROB] FRAC / 1:rand / 2:nPr / 3:nCr / 4:! / 5:randInt( / 6:randNorm( / 7:randBin(.
The PROB tab.
ItemWhat it gives
1:randA random number between 0 and 1. Store a number first (for example 7→rand) to start from a fixed seed.
2:nPrNumber of ordered arrangements: 5 nPr 2 = 20.
3:nCrNumber of selections (combinations): 5 nCr 2 = 10.
4:!Factorial: 5! = 120.
5:randInt(A random whole number between two limits, with an optional count: randInt(1,6).
6:randNorm(A random number from a normal distribution: randNorm(mean, sd).
7:randBin(A random count from a binomial distribution: randBin(trials, p).
nPr=n!(n−r)!,nCr=n!r! (n−r)!{}_nP_r=\frac{n!}{(n-r)!},\qquad {}_nC_r=\frac{n!}{r!\,(n-r)!}
Permutations and combinations.
Example 4.21

Combinations

  1. Type 5, press math, left, left, 3 for nCr and type 2.

    5math◀◀32enter
Result of the example "Combinations" The display shows: 5 nCr 2 / 10.
You should see5 nCr 2 10
Example 4.22

Permutations

  1. Type 5, then the PROB item 2 (nPr) and 2.

    5math◀◀22enter
Result of the example "Permutations" The display shows: 5 nPr 2 / 20.
You should see5 nPr 2 20
Example 4.23

Factorial

  1. Type 5, then PROB item 4.

    5math◀◀4enter
Result of the example "Factorial" The display shows: 5! / 120.
You should see5! 120
Example 4.24

A random whole number

  1. Choose randInt( and type 4, 4. When both limits are equal the answer is that number, so you can see the pattern of the command.

    math◀◀54,4)enter
Result of the example "A random whole number" The display shows: randInt(4,4) / 4.
You should seerandInt(4,4) 4

4.6Compare numbers and use logic (TEST menu)#

Press 2ndTESTmath for TEST. A comparison gives 1 when it is true and 0 when it is false. Press ▶ for the LOGIC tab.

TabItems
TEST1: = 2: ≠ 3: > 4: ≥ 5: < 6: ≤
LOGIC1: and 2: or 3: xor 4: not(
The TEST menu. The display shows: [TEST] LOGIC / 1:= / 2:≠ / 3:> / 4:≥ / 5:< / 6:≤.
The TEST menu.
Example 4.25

Is 3 equal to 3?

  1. Type 3, press 2nd math then 1 for the equals sign, type 3.

    32ndTESTmath13enter
Result of the example "Is 3 equal to 3?" The display shows: 3=3 / 1.
You should see3=3 1
Example 4.26

Is 2 greater than 3?

  1. Type 2, 2nd math, 3 for >, then 3.

    22ndTESTmath33enter
Result of the example "Is 2 greater than 3?" The display shows: 2>3 / 0.
You should see2>3 0
Example 4.27

Combine with and

  1. Type 1, press 2nd math, right, 1 for and, then 0.

    12ndTESTmath▶10enter
Result of the example "Combine with and" The display shows: 1 and 0 / 0.
You should see1 and 0 0
Example 4.28

Not

  1. Press 2nd math, right, 4 for not( and type 0.

    2ndTESTmath▶40)enter
Result of the example "Not" The display shows: not(0) / 1.
You should seenot(0) 1

4.7Constants: π, e and i#

ConstantKeyValue
π2ndπ^3.141592654
e2nde÷2.718281828
i2ndi.The imaginary unit, √(−1). Needs a complex mode.
Example 4.29

Show π

  1. Press 2nd then the ^ key.

    2ndπ^enter
Result of the example "Show π" The display shows: π / 3.141592654.
You should see3.141592654
Example 4.30

Show e

  1. Press 2nd then the ÷ key.

    2nde÷enter
Result of the example "Show e" The display shows: e / 2.718281828.
You should see2.718281828

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