Chapter 2: Math, Angle, and Test Operations

Using Lists with Math Operationsp. 36

Any math operation that accepts lists works element by element and returns a list. If you use two lists in the same expression they must have the same length, otherwise the calculator reports a dimension error.

Addition, Subtraction, Multiplication, Divisionp. 36

The four arithmetic keys work with real and complex numbers, expressions, lists and matrices. The one exception is division: you cannot divide by a matrix. To enter A/2 for a matrix A, write it as A × 0.5 or as A multiplied by the fraction 1/2.

Trigonometric Functionsp. 36

Sine, cosine and tangent (and their inverses) accept real numbers, expressions and lists. They do not accept complex numbers. How the argument is read, or what angle is returned, depends on the current angle mode (Radian or Degree).

Power, Square, Square Rootp. 36

The power, square and square-root keys work with real and complex numbers, expressions, lists and matrices. The one exception is the square root, which does not accept matrices. In MathPrint mode, pressing ^ opens a raised exponent slot; type the exponent and press ▶︎ to get back to the baseline. In Classic mode the expression is written value^power.

Inversep. 37

The x⁻¹ key gives the multiplicative inverse, which is the same as the reciprocal 1/x. It works with real and complex numbers, expressions, lists and square matrices.

log(, 10^(, ln(p. 37

The logarithm keys work with real or complex numbers, expressions and lists. log( is the base-10 logarithm, 10^( is its inverse, and ln( is the natural (base e) logarithm.

Exponentialp. 37

Constant (e)p. 37

The constant e is stored on the calculator. Press 2nde to paste it at the cursor. In calculations the calculator uses 2.718281828459 for e.

Negationp. 37

The (−) key ((−)) makes a number negative. It works with real or complex numbers, expressions, lists and matrices. Negation follows the EOS order-of-operations rules from Chapter 1: squaring happens before negation, so ⁻4² is −16. To square a negative number, put it in parentheses: (⁻4)² is 16.

Pip. 38

π is a stored constant. Press 2ndπ to paste it. In calculations the calculator uses 3.1415926535898 for π.

►Frac, ►Decp. 39

►Frac shows an answer as the equivalent fraction. It works with real or complex numbers, expressions, lists and matrices. If the answer cannot be written as a simple fraction, or the denominator would have more than three digits, you get the decimal back instead. ►Dec does the reverse and shows an answer in decimal form. Both can only be used after a value, at the end of an expression.

Cube, Cube Rootp. 39

³ cubes a value (real or complex numbers, expressions, lists and square matrices). ³√( takes the cube root of real or complex numbers, expressions and lists.

ˣ√ (Root)p. 40

ˣ√ returns the xth root of a value. It works with real or complex numbers, expressions and lists. The root number goes first, and the value you are taking the root of comes after the radical sign.

fMin(, fMax(p. 40

fMin( and fMax( search an interval for the local minimum or maximum of an expression. They return the value of the variable at which that extreme occurs, not the extreme value itself. You give the expression, the variable, and the lower and upper ends of the interval. The optional tolerance controls accuracy; the default is 1E⁻5.

nDeriv(p. 40

nDeriv( returns an approximate derivative of an expression at a point. It uses the symmetric difference quotient: the slope of the secant line through the points at x−ε and x+ε,

f′(x) ≈ ( f(x+ε) − f(x−ε) ) / (2ε)

where ε is the optional step H (default 1E⁻3). A smaller H usually gives a more accurate answer. In MathPrint mode the template has no slot for H, so switch to Classic mode if you want to experiment with it. nDeriv( works only with real numbers, and it can be used only once in an expression.

fnInt(p. 41

fnInt( computes a numerical integral with the Gauss-Kronrod method. You give the expression, the variable, and the lower and upper limits; the optional tolerance defaults to 1E⁻5. It works only with real numbers. In MathPrint mode it is the ∫ template, with the limits entered first.

summation Σ(p. 39

summation Σ( adds up an expression as a variable runs from a start value to an end value (start ≤ end). In MathPrint mode it is the Σ template: the variable and its start value go below the sigma, the end value above, and the expression to the right.

logBASE(p. 39

logBASE( gives the logarithm of a value in any base you choose. The function is written logBASE(value,base). In MathPrint mode the template asks for the base first, in the small subscript slot, then the value.

Using the Equation Solverp. 42

Solver lets you solve for any variable in an equation. The calculator treats your equation as equal to zero, and it works only with real numbers. When you choose B:Solver... one of two screens appears:

• the equation editor, if the equation variable eqn is empty;
• the interactive solver editor, if an equation is already stored in eqn.

abs(p. 46

abs( returns the absolute value of a real number, or the modulus of a complex number, and works with expressions, lists and matrices. It is also on the MATH CPX menu and on the FUNC shortcut menu (ALPHAF2 then 1).

round(p. 47

round( rounds a number, expression, list or matrix to a number of decimal places (up to 9). If you leave out the number of decimals, the value is rounded to the digits currently displayed, up to 10 digits.

iPart(, fPart(p. 47

iPart( returns the integer part of a number (it just drops the digits after the decimal point), and fPart( returns what is left: the fractional part. Both work with real or complex numbers, expressions, lists and matrices.

int(p. 47

int( returns the largest integer that is less than or equal to the value (the "floor"). It works with real or complex numbers, expressions, lists and matrices.

min(, max(p. 48

min( returns the smaller of two values, or the smallest element of a list. Given two lists it compares them position by position and returns a list of the smaller of each pair. Given a list and a single value, it compares every element with that value. max( does the same with the larger value.

lcm(, gcd(p. 48

lcm( returns the least common multiple and gcd( the greatest common divisor of two non-negative integers. With two lists they work pair by pair and return a list; with a list and a value, they work on each list element together with the value.

remainder(p. 49

remainder( gives the remainder when one whole number is divided by another (dividend and divisor). Either argument can be a list. The divisor cannot be zero. If both are lists they must be the same length; if only one is a list, the single value is paired with every list element and a list is returned.

►n/d◄►Un/dp. 49

►n/d◄►Un/d converts an improper fraction to a mixed number, or a mixed number to an improper fraction. It is also item 3 on the FRAC shortcut menu (ALPHAF1).

►F◄►Dp. 50

►F◄►D converts a fraction to a decimal, or a decimal to a fraction. It is also item 4 on the FRAC shortcut menu (ALPHAF1).

Un/dp. 50

Un/d puts the mixed-number template on the screen (item 2 of the FRAC shortcut menu). In MathPrint mode it is a whole number with a fraction beside it; in Classic mode a small u separates the whole number from the fraction. In a mixed number, n and d must be non-negative integers.

n/dp. 50

n/d puts the fraction template on the screen (item 1 of the FRAC shortcut menu). Here n and d can be real numbers or expressions, but they may not contain complex numbers. In Classic mode the fraction is shown with a thick fraction bar.

Complex-Number Modesp. 50

Press MODE and pick one of the two complex-number modes on the Real / a+bi / re^θi line:

• a+bi: rectangular-complex mode.
• re^θi: polar-complex mode.

In Real mode, a complex result produces an error unless one of the numbers you entered was complex. For example, ln(⁻1) gives ERR:NONREAL ANS in Real mode but returns an answer in a+bi mode.

Entering Complex Numbersp. 51

Complex numbers are stored in rectangular form, but you can type one in either rectangular or polar form regardless of the mode setting. The real and imaginary parts can be real numbers or expressions that evaluate to real numbers; expressions are evaluated when you press ENTER. The imaginary unit i is 2ndi.

You can use fractions in a complex number, but the output will always be a decimal. When you use the n/d template, the fraction itself cannot contain a complex number, so use division to compute such an answer instead.

Note about Radian Versus Degree Modep. 52

Use Radian mode for complex-number work. The calculator converts trigonometric arguments to radians internally, but it does not convert the arguments of exponential, logarithmic or hyperbolic functions.

In Degree mode this breaks identities such as e^(iθ) = cos(θ) + i·sin(θ). For instance, e^(i45) = cos(45) + i·sin(45) is evaluated as e^(i45) = cos(π/4) + i·sin(π/4), so the two sides no longer match. In Radian mode complex identities always hold.

Interpreting Complex Resultsp. 52

A complex number in a result (including a list element) is shown in rectangular or polar form, according to the mode setting or according to a display-conversion instruction such as ►Rect or ►Polar. Polar-complex mode is easiest to see in Radian mode.

conj(p. 54

conj( returns the complex conjugate of a complex number or a list of complex numbers. In a+bi mode, conj(a+bi) returns a−bi. In re^θi mode, conj(re^(θi)) returns re^(⁻θi).

real(p. 54

real( returns the real part of a complex number or list of complex numbers. real(a+bi) returns a; real(re^(θi)) returns r·cos(θ).

imag(p. 55

imag( returns the imaginary (non-real) part of a complex number or list of complex numbers. imag(a+bi) returns b; imag(re^(θi)) returns r·sin(θ).

angle(p. 55

angle( returns the polar angle of a complex number or list of complex numbers. It is calculated as tan⁻¹(b/a), where b is the imaginary part and a the real part, then adjusted by +π in the second quadrant or −π in the third quadrant. angle(re^(θi)) returns θ, with −π < θ < π. The answer is in the current angle mode.

abs( (complex)p. 55

abs( returns the magnitude (modulus) √(a²+b²) of a complex number or list of complex numbers. abs(a+bi) returns √(a²+b²) and abs(re^(θi)) returns r. It is the same function as the one on the NUM tab and is also on the FUNC shortcut menu (ALPHAF2 then 1).

►Rectp. 56

►Rect shows a complex result in rectangular form (a+bi). It is valid only at the end of an expression, and it is not valid if the result is real.

►Polarp. 56

►Polar shows a complex result in polar form re^(θi). It is valid only at the end of an expression, and it is not valid if the result is real.

randp. 57

rand generates one or more random numbers greater than 0 and less than 1. To get a list of random numbers, give a whole number greater than 1 as the number of trials; the default is 1.

For a given seed value, every rand run produces the same sequence of numbers. The factory-set seed is 0. To get a different sequence, store any non-zero number into rand (for example 123→rand). Storing 0 into rand, or resetting the defaults (Chapter 18), restores the factory seed. The seed also affects randInt(, randNorm( and randBin(.

nPr, nCrp. 57

nPr counts the permutations (ordered selections) of items taken number at a time. nCr counts the combinations (unordered selections). Both arguments must be non-negative integers, and either can be a list.

! (Factorial)p. 58

! returns the factorial of an integer or of a multiple of .5. For a list, it returns the factorial of each element. The value must be ≥ ⁻.5 and ≤ 69.

The calculator computes the factorial step by step using (n+1)! = (n+1)·n!, until n is reduced to 0 or ⁻1/2. It then uses the definitions 0! = 1 and (⁻1/2)! = √π. Therefore:

• n! = n·(n−1)·(n−2)·…·2·1, if n is an integer ≥ 0;
• n! = n·(n−1)·…·(1/2)·√π, if n+1/2 is an integer ≥ 0;
• n! is an error if neither n nor n+1/2 is an integer ≥ 0.

randInt(p. 58

randInt( generates a random integer between a lower and an upper whole-number bound (both included). Add a third argument greater than 1 to get a list of that many random integers; the default is 1.

randNorm(p. 58

randNorm( generates a random real number from a Normal distribution with mean μ and standard deviation σ. Any real number is possible, but most results fall within three standard deviations of the mean, in the interval [μ−3σ, μ+3σ]. A third argument greater than 1 gives a list; the default is 1.

randBin(p. 59

randBin( generates a random integer from a Binomial distribution: the number of successes in numtrials independent trials, each with success probability prob. numtrials must be at least 1, and prob must be between 0 and 1. A third argument (the number of simulations) greater than 1 gives a list; the default is 1. The Coin Flip activity at the start of the chapter uses it.

randIntNoRep(p. 59

randIntNoRep( returns a list of the integers from a lower integer to an upper integer, each appearing exactly once, in random order. Use it to shuffle or to sample without repeats.

Entry Notation (DMS)p. 60

DMS (degrees/minutes/seconds) entry notation uses three symbols: degrees (°), minutes (') and seconds ("). The degrees must be a real number; minutes and seconds must be real numbers ≥ 0. Fractions are not supported in the minutes or seconds.

degrees°minutes'seconds"

As a check, 30 degrees is the same as π/6 radians. You can verify this by evaluating 30° in Radian mode (which converts it to radians) and comparing it with π/6, or by looking at the sine in each mode.

° (Degree)p. 60

° marks a value (or every element of a list) as degrees, whatever the current angle mode is. In Radian mode, use ° to convert degrees to radians. In DMS format it also stands for the degrees part.

value° or {value1,value2,…}°

ʳ (Radians)p. 61

ʳ marks a value or list as radians whatever the current angle mode is. In Degree mode, use ʳ to treat a number as radians when you apply a trig function.

valueʳ

►DMSp. 61

►DMS displays an answer in degrees/minutes/seconds. The mode must be Degree for the answer to be read as degrees, minutes and seconds. ►DMS is valid only at the end of a line.

R►Pr(, R►Pθ(, P►Rx(, P►Ry(p. 61

R►Pr( and R►Pθ( convert rectangular coordinates (x, y) to polar coordinates, returning r and θ. P►Rx( and P►Ry( convert polar coordinates (r, θ) to rectangular coordinates, returning x and y. The arguments can be lists. Angles are read or returned in the current angle mode.

Using Testsp. 62

Relational operators are evaluated after the mathematical functions, following the EOS rules from Chapter 1.

• 2+2=2+3 returns 0. The additions happen first, and then 4 is compared with 5.
• 2+(2=2)+3 returns 6. The parentheses force the test to happen first (it is true, so 1), and then 2 + 1 + 3 is added.

and, or, xorp. 63

and, or and xor (exclusive or) return 1 when their condition is true and 0 when it is false, according to the truth table below. valueA and valueB can be real numbers, expressions or lists.

not(p. 63

not( returns 1 if the value (which can be an expression) is 0, and 0 otherwise.

Using Boolean Operationsp. 63

Boolean logic is often combined with relational tests. In the program shown in the guidebook, such a test decides whether the instructions store 4 into C. You can try the same idea right on the home screen, since a true test is just the number 1 and a false test is 0.