1. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Prof. Gevelyn B. Itao

2. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Decimal system – base 10 Base 5 system Binary system – base 2 Hexadecimal system – base 16 Octal system – base 8

3. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Decimal or base-10: 10 different digits for counting, i.e., 0,1,2,3,4,5,6,7,8,9. (Human and natural, how many fingers do you have?). Binary or base-2: only 2 digits for counting, i.e., 0 and 1. (All the computers use binary number system.)

4. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology It is easy and natural to realize a “0” and “1” by a hardware device that has only two states – “off” and “on”. A typical such device is a transistor. You can think of a transistor as a light bulb; it can either be bright when it is turned “on” or dark when it is turned “off”. So a 8-bit binary number is represented by 8 light bulbs in a computer. 1 0 1 0 1 1 0 1

5. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology In an arbitrary number base b, an m-digit integer I m is written as: Where b is the number base, and the digits d j is the set of integers from 0 to (b-1). The decimal value of an integer above is obtained from

6. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology The largest number that can be stored in m digits is one in which every digit is equal to (b-1). The decimal equivalent of this number is

7. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology (- 1) Sign ×Number Memory cells usually consists of 8-bit (1-byte) multiples, with each position containing one binary digit or bit. 1. Sign-magnitude notation 2. Two’s complement notation Two’s complement of a negative integer

8. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology An n-digit fractional part P n may be written in a number base b as Floating point numbers or reals contain an integer part and a fractional part. Obtaining the decimal value of P n is

9. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology An n-digit fractional part P n may be written in a number base b as Obtaining the decimal value of P n is

10. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology (-1) Sign ×Mantissa×b Exponent (b is the base of the number system used) Mantissa is usually required to be normalized: ≤ Mantissa < 1 Example (b = 10): 0.0294 = 0.294×10 -1

11. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Convert the following decimal numbers to the designated number systems and determine the percent relative round-off error 1. (39.99) 10 to binary number system in 10 significant bits in the fractional part 2. (87.12) 10 to hexadecimal number system in 10 significant hexadecimal digits in the fractional part 3. (0.236) 10 to octal number system in 10 significant octal digits

12. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology Convert the following decimal numbers to the designated number systems and determine the percent relative round-off error 1. (0.99) 10 to binary number system in 8 significant bits 2. (1.6666) 10 to octal number system in 10 significant octal digits in the fractional part

13. Algebraic, transcendental (i.e., involving trigonometric and exponential functions), ordinary differential equations, or partial differential equations... ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

14. :::: : Where x j (j=1,2,…m) denotes the unknown variable a ij (i=1,2,…n; j=1,2,…m) denotes the coefficients of the unknown variable b i (i=1,2,…n) denotes the non-homogeneous terms

15. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology  One that produces either exact or approximate solutions in closed form.

16. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1. A unique solution – a consistent set of solutions 2. No solution – an inconsistent set of equations 3. An infinite number of solutions – a redundant set of equations 4. The trival solution x j = 0 – a set of homogenous equations

17. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology 1. Direct methods – are systematic procedures, based on algebraic elimination, that obtain the solution in a fixed number of operations. 2. Iterative methods – obtain the solution asymptotically by an iterative procedure. A trial solution is assumed, the trial solution is substituted into the system of equations to determine the mismatch in the trial solution, and an improved solution is obtained from the mismatch data.