# Casio fx-3650P Programming Manual

Programmable scientific calculator

• ### Page 4: Programming A Calculation

Basic Operation for fx-3650P/3950P Programming a Calculation PRGM • The program storage area has about 360 bytes, which can be 2. Input the values required for the calculation. divided among up to four different programs named P1, P2, P3, A? 30 and P4.
• ### Page 5: Greatest Common Divisors

Greatest Common Divisors Given two natural numbers, , the greatest common divisor may be found using Euclid’s algorithm. A simplified explanation is given below: > be two natural numbers. １．Let be the remainder after dividing ２．Let ≠ → → , set and return to 2.
• ### Page 6: Fractions From Recurring Decimals

Fractions from recurring decimals Recurring decimals may be converted to fraction by proceeding in the following way: …… …… 0.111111111111 ----- - 0.01010101010 -- - …… …… -------- - 0.001001001001001 ----------- - 0.000100010001 9999 For example, 0.345345345…… 0.345345345 …… -------- - -------- - PRGM COMP...
• ### Page 7: Simultaneous Equations

Simultaneous Equations ≠ AE BD – When , the solution to the simultaneous equation:    is given by the formula: EC FB – AF CD – -------------------- - --------------------- - PRGM COMP MODE MODE MODE MODE Ｌｂｌ　１：？→ Ａ：？→ Ｂ：？→ Ｃ：Ａ → Ｙ：Ｂ → Ｘ：Ｃ → Ｍ：？→ Ａ：？→ Ｂ：？→ Ｃ：ＡＸ...
• Page 8 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 AE BD – When , we cannot guarantee that there is a unique solution. In this case please enter a different problem.
• ### Page 9: Solutions To Quadratic Equations

Solutions to quadratic equations The solutions to the quadratic equation are given by the formula: ± – – -------------------------------------- – > In particular, when two real solutions exist, – when there is just one real solution and – < when there are two complex solutions PRGM CMPLX...
• Page 10 ± – ii) Find solutions to the equation Prog CMPLX P1 P1 P2 P3 P4 CMPLX P1 P1 P2 P3 P4 <> Disp SHIHT CMPLX P1 P1 P2 P3 P4 <> Disp CMPLX P1 P1 P2 P3 P4 <> SHIHT P1 P1 P2 P3 P4 CMPLX <>...
• ### Page 11: Values Of Functions

Values of Functions For two variables where the values of determined for given values of , we say that is a function of For example, for the function – when , and 3 – 1 – when PRGM COMP MODE MODE MODE MODE...
• ### Page 12: Summing Arithmetic Progressions

Summing Arithmetic Progressions １． The sum up to the th term of an arithmetic progression with first term difference term , may be calculated by: … ∑ – – --- - 2A – PRGM COMP MODE MODE MODE MODE ？→ Ａ：？→ Ｂ：？→ Ｃ：Ｃ（２ Ａ ＋（Ｃ －１）Ｂ）÷２→ Ｄ：Ｄ　＜ 30 STEP ＞ OUTPUT Ｄ: sum of arithmetical progression Calculate the sum up to the 10th term of the arithmetic progression with first term 1 and...
• Page 13 ２． The sum of an arithmetic progression with first term and final term , consisting of terms is given by –   … ------------ -   – terms PRGM COMP MODE MODE MODE MODE ？→ Ａ：？→ Ｂ：？→ Ｃ： （Ａ ＋ Ｂ）Ｃ ÷２→ Ｄ：Ｄ　＜ 24 STEP ＞ OUTPUT Ｄ: sum of arithmetical progression The of an arithmetic progression of 10th terms, with first term 1 and last term 28, is:...
• ### Page 14: Summing Geometric Progressions

Summing Geometric Progressions １． The sum up to the th term of the geometric progression with first term and ratio term may be calculated: – … – ∑ A 1 B – ----------------------- - – PRGM COMP MODE MODE MODE MODE ？→...
• Page 15 ２． The sum of a geometric progression with first term , last term and common ratio A BC – … ⁄ ---------------- - – PRGM COMP MODE MODE MODE MODE ？→ Ａ：？→ Ｂ：？→ Ｃ： （Ａ － ＢＣ）÷（１－ Ｃ）→ Ｄ：Ｄ　＜ 28 STEP ＞ OUTPUT Ｄ: sum of geometric progression The sum of the geometric progression with first term 1, last term 81 and common ratio 3 is...
• ### Page 16: Approximate Solutions To Higher Order Equations(Bisection Method)

Approximate solutions to higher order equations (bisection method) f x ( ) be a function continuous on the interval f a ( ) f b ( ) [ , ] , with having opposite signs, so [ , ] that there must be at least one with f c ( ) If we divide the interval into two,...
• Page 17 P1 P1 P2 P3 P4 Disp To stop program, press the key. To calculate for other functions, alter the program where underlined.
• ### Page 18: Approximate Solutions To Equations Of Higher Degree (Newton's Method)

Approximate solutions to Equations of higher degree (Newton’s method) f x ( ) Newton’s method is used to calculate approximate solutions to the equation . For a initial value , the following recurrence relation gives successive approximate solutions: – ------------ ′...
• ### Page 19: Limits

Limits   -- - The value of the limit is the base of   ∞ → the natural logarithm, namely PRGM COMP MODE MODE MODE MODE ２ Ｘ → Ｘ：Ｇｏｔｏ　１　＜ 28 STEP ＞ １→ Ｘ：Ｌｂｌ　１： （１＋１÷ Ｘ） ＾ Ｘ → Ｙ：Ｙ ...
• ### Page 20: Limits

Limits --------- - It is known that the limit takes value 1. → PRGM COMP MODE MODE MODE MODE Ｘ ÷２→ Ｘ：Ｇｏｔｏ　１　＜ 24 STEP ＞ １→ Ｘ：Ｌｂｌ　１：ｓｉｎ　Ｘ ÷ Ｘ → Ｙ：Ｙ --------- - Check the convergence of for values of close to 0.
• ### Page 21: Definite Integrals(Trapezoidal Rule)

Definite Integrals (Trapezoidal rule) Approximations of the value of definite integrals may be obtained as follows: … ∫ x ( ) x d ≈ -- - f A ( ) f B ( ) 2f A 2f A 2f B h –...
• Page 22 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 For large n, the calculation may take a considerable amount of time. To calculate for other functions, alter the program where underlined.
• ### Page 23: Definite Integrals(Simpsons Formula)

Definite integrals (Simpsons formula) The value of a definite integral can be approximated by Simpsons formula,   –   ∫ ∑ ∑ f x ( ) x d ≈ -- - f A ( ) f B ( ) 2i 1 –...
• Page 24 P1 P1 P2 P3 P4 For large n, the calculation may take a considerable amount of time. To calculate for other functions, alter the program where underlined.
• ### Page 25: Finding The Area Of A Triangle

Finding the area of a triangle In general, when the length of the base and the height of a triangle are known, its area can be calculated by （Length of base）×（Height）÷２ . However, it is also possible to determine the area given the lengths of the three sides, using the following formula. Let the lengths of the three sides be ，...
• ### Page 26: The Angle Between Two Vectors

The angle between two vectors θ → The angle between two vectors → , not equal to 0 for 0 being the origin, can be obtained in the following way: → cos a b ⋅ → θ ----------- - 1 – →...
• ### Page 27: Finding The Angles In A Triangle

Finding the angles in a triangle Law of cosines：The relationship between the length of the sides of a triangle and the angle between is given by: θ – θ Solving this for we obtain the angle in terms of the lengths: cos b –...
• ### Page 28: The Radius Of The Inscribed Circle

The radius of the Inscribed circle. The inscribed circle is shown in the diagram on the right.When the inscribed circle has radius , the area of the appropriate triangle is: A B C ----- - ----- - ----- - ----------------------------- Now, using herons formula it is possible to determine the area of the triangle in terms of the length of the three sides, whereby the radius of the circle is:...
• ### Page 29: The Radius Of The Circumscribed Circle

The radius of the Circumscribed circle The law of sines：For △ , as shown in the diagram on the right, the following formula holds: ---------- - ---------- - ----------- It is thereby possible, by the law of cosines and law of sines, to calculate the radius of the circumscribed circle given the lengths of the three sides of the triangle.
• ### Page 30: Center Of Mass Of A Triangle(Barycentre)

Center of mass of a triangle (barycentre) The three medians of a triangle all meet at a single point. This point is called the “center of mass” (or barycentre). The center of mass divides each line in half. For a triangle with corner points the center of mass is given: -------------------------------------- - PRGM...
• ### Page 31: The Distance Between A Line And A Point

The distance between a line and a point ( , ) The distance between a point and a straight line (x, y) , as in the diagram on the right, is given by the formula: y=ax+b ax y – ----------------------- - PRGM COMP MODE...
• ### Page 32: The Sum, Difference And Inner Product Of Two Vectors

The sum, difference and inner product of two vectors The sum and difference of two vectors in 3-dimensional space is obtained by adding and subtracting their respective components. → α β → ′ ′ c′ For example, for → α β →...
• Page 33 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 Disp P1 P1 P2 P3 P4 &&...
• ### Page 34: Rotation

Rotation x y , The following matrix rotates a point , in the plane, θ through an angle of in the anti-clockwise direction. (x’, y’)       θ θ x′ –   (x, y)  ...
• ### Page 35: Complex Numbers And Polar Coordinates

Complex numbers and polar coordinates ω As an example, for the complex number ω we may write in polar coordinates, by letting ω θ θ a + bi θ θ -- - -- - where The product of two complex numbers in polar form is θ...
• ### Page 36: The Inverse Of A 2 × 2 Matrix

The inverse of a 2 × 2 matrix.     b – ----------------- -     The inverse of the ２×２ matrix . In other words, ad bc –    c – a  ...
• ### Page 37: Combinations

Combinations Let there be cards with the letter “ cards with the ”, letter “ ” and cards with the letter “ ” written on them. Then number of possible different ways to arrange all × × cards in a row, is given by: ---------------------------- - b ×...
• ### Page 38: The Binomial Distribution

The binomial distribution Assume that the probability that a tossed coin will land showing “heads” is , and that the – probability for “tails” is ( ). Then over tosses, the probability that the coin will land times with “heads”-up, is: –...
• ### Page 39: Confidence Intervals

Confidence Intervals σ For a random sample of size from normal distribution with known variance , the µ confidence intervals for the population mean may be calculated as given by: σ σ µ × × ------ - ------ - x 1.64 –...
• Page 40 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 P1 P1 P2 P3 P4 Prog P1 P1 P1 P1 P2 P3 P4 P1 P1 P1 P1 P2 P3 P4 Disp P1 P1 P1 P1 P2 P3 P4 Disp...
• ### Page 41: Test On A Mean

σ µ Test on a mean with known (z-tests) µ The hypothesis of interest is that the mean is equal to a given value , versus the µ alternative that it is not equal to . We calculate the test statistics µ...
• Page 42 Answer 1.442857143 µ – = 1.442857143－ 1.2 1.4428571 1.2111 – ----------- - ------------------------------------------------ - 4.543441113 σ 0.02 ------ - ------------- - α ≥ 1.96 0.05 , we reject the hypothesis for the significance level P1 P1 P2 P3 P4 MODE MODE P1 P1 P2 P3 P4 P1 P1 P2 P3 P4...
• ### Page 43: Guess The Number

Guess the number Assume you wish to guess a randomly chosen number between 1 and 1000. Simple guessing will require, on average, 500 tries to find the answer. This program will tell you if your guess is larger or smaller than a number (which it chooses), so that, proceeding logically, it should be possible to find the answer within 11 attempts.
• ### Page 44: Various Areas And Volumes

Various areas and volumes Area: OUTPUT Ｍ: areas. Triangle： （Base　 Ａ Ｂ ）×（Height　 ）÷２ ？→ Ａ：？→ Ｂ：ＡＢ ÷２→ Ｍ：Ｍ ２ Square： （Length of sides　 Ａ ） ２ ？→ Ａ：Ａ → Ｍ：Ｍ Rectangle： （Height　 Ａ Ｂ ）×（Width　 ） ？→ Ａ：？→ Ｂ：ＡＢ → Ｍ：Ｍ Trapezium：...
• Page 45 Volume: OUTPUT Ｍ: volumes ３ Cube： （Length of sides　 Ａ ） ３ ？→ Ａ：Ａ → Ｍ：Ｍ Rectangular prism： （Depth　 Ａ Ｂ Ｃ ）×（Width　 ）×（Height　 ） ？→ Ａ：？→ Ｂ：？→ Ｃ：ＡＢＣ → Ｍ：Ｍ Triangular, Quadrangular pyramid：(Base Area ) × (Height ) ÷３ ？→...
• Page 46 MEMO MEMO MEMO MEMO ● ●...
• Page 47 Authors Dr. Yuichi Takeda Research and Development Initiative Chuo University...
• Page 48 MI0209-01001 Printed in Japan...

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