Miscellaneous

Including: Vectors, HYP Functions, Complex & Real Numbers.

Vectors

1) ABS

Calculates the norm of a vectror

The norm of a vector **[ a b c ]** is definite as **SQRT(a^2 + b^2 + c^2)**

Example::

Find the norm of the vector [2 3 4]

level 1: **[2 3 4]**

[MTH][VECTR][ABS]

**Result: 5.38516480713**

2) DOT

Calculates the scalar product of two vectors.

The result is a number.

Example::

**[2 3 4].[6 7 8]**

level 2: **[2 3 4]**

level 1:** [6 7 8]**

[MTH][VECTR][DOT]

**Result: 65**

3) CROSS

Calculates the vectorial produc of two vectors.

The result is a vector.

Example::

**[2 3 4] x [6 7 8]**

level 2:** [2 3 4]**

level 1:**[6 7 8]**

[MTH][VECTR][CROSS]

**Result: [-4 8 -4]**

4) V2

Convert numbers in levels 1 and 2 into a vector.

Example::

level 2:** 9
**level 1:

[MTH][VECTR][V2]

5) V3

Convert numbers in levels 1, 2 and 3 into a vector.

Example::

level 3:

level 2:

level 1:

[MTH][VECTR][V3]

6) V

Convert a vector into stack numbers.

Example::

level 1:

[MTH][VECTR][V]

level 3:

level 2:

level 1:

Hyperbolic functions are functions based on **exp(x)**

To find the value of an hyperbolic trigonometric

funtion all you need is to put the number

on the stack and press the respective function at [MTH][HYP]

Example::

Calculate the**sinh(5)**

level 1:**5 **

[MTH][HYPER][SINH]

**Result: 74.2032105778 **

funtion all you need is to put the number

on the stack and press the respective function at [MTH][HYP]

Example::

Calculate the

level 1:

[MTH][HYPER][SINH]

In HP48/49 complexes numbers are represented between **( )**.

as (real part, complex part)

Work with complexes numbers in HP48/49 is easy.

Below are listed the most useful functions.

1) RE

Returns the real part of a complex number

Example::

Find the real part of**(123, 456)**

Level 1:**(123, 456) **

Press [MTH][NXT][CMPL][RE]

**Result: 123 **

2) IM

Returns the imaginary part of a complex number

Example::

Find the imaginary part of**(123, 456)**

Level 1:**(123, 456)**

Press [MTH][NXT][CMPL][IM]

**Result: 456 **

3) ABS

Returns the norma of a complex number

Example::

Find the norm of the complex number**(123, 456)**

Level 1:**(123, 456)**

Press [MTH][NXT][CMPL][ABS]

**Result: 472.297575687**

4) ARG

Return the angle argument of a complex number

Example::

Find the angle argument of**(123, 456)**

Level 1:**(123, 456)**

Press [MTH][NXT][CMPL][ARG]

**Result:**

74.9044791558 (angle mode)

1.30732978576 (RAD mode)

5) CONJ

Return the conjugate of a complex number

Example::

Find the conjugate of**(123, 456)**

Level 1:**(123, 456)**

Press [MTH][NXT][CMPL][ARG]

**Result: (123, -456)**

as (real part, complex part)

Work with complexes numbers in HP48/49 is easy.

Below are listed the most useful functions.

1) RE

Returns the real part of a complex number

Example::

Find the real part of

Level 1:

Press [MTH][NXT][CMPL][RE]

2) IM

Returns the imaginary part of a complex number

Example::

Find the imaginary part of

Level 1:

Press [MTH][NXT][CMPL][IM]

3) ABS

Returns the norma of a complex number

Example::

Find the norm of the complex number

Level 1:

Press [MTH][NXT][CMPL][ABS]

4) ARG

Return the angle argument of a complex number

Example::

Find the angle argument of

Level 1:

Press [MTH][NXT][CMPL][ARG]

74.9044791558 (angle mode)

1.30732978576 (RAD mode)

5) CONJ

Return the conjugate of a complex number

Example::

Find the conjugate of

Level 1:

Press [MTH][NXT][CMPL][ARG]

1) RND

Rounds a number for a definite numbers of digits

Level 2:**1.8678987**

Level 1:**3**

Press [MTH][REAL][RDN]

**Result: 1.868 **

2) TRNC

Truncates a number to the set numbers of digits

Level 2:**1.8678987**

Level 1:**3**

Press [MTH][REAL][TRNC]

**Result: 1.867 **

3) MOD

Returns the rest of a division between two numbers

Example::

Level 2:**14 **

Level 1:**3**

Press [MTH][REAL][MOD]

**Result: 2 **

**Note: 8 / 3 = 4 and rest 2. **

4) IP

Show the Integer Part of a number

Example::

Level 2:**1.8678987**

Press [MTH][REAL][IP]

**Result: 1 **

5) FP

Show the Fraction Part of a number

Example:

Level 2:**1.8678987**

Press [MTH][REAL][FP]

**Result:. 8678987 **

6) DR

Converts from Degree to radians

Example:

Level 2:**45**

Press [MTH][REAL][DR]

Result:**.785398163397**

7) RD

Converts from Radians to Degree

Example:

Level 2:**.785398163397 Example:**

Press [MTH][REAL][RD]

**Result: 45 **

8) CEIL

Returns the mayor integer closest to the number

Example:

Level 2:**1,45 **

Press [MTH][REAL][CEIL]

**Result: 2 **

9) FLOOR

Returns the minor integer closest to the number

Example:

Level 2:**1,45 **

Press [MTH][REAL][FLOOR]

**Result: 1 **

10) %

Returns the percentage

Example:

Level 2:**100**

Level 1:**12,67**

Press [MTH][REAL][%]

**Result:12.67 **

Rounds a number for a definite numbers of digits

Level 2:

Level 1:

Press [MTH][REAL][RDN]

2) TRNC

Truncates a number to the set numbers of digits

Level 2:

Level 1:

Press [MTH][REAL][TRNC]

3) MOD

Returns the rest of a division between two numbers

Example::

Level 2:

Level 1:

Press [MTH][REAL][MOD]

4) IP

Show the Integer Part of a number

Example::

Level 2:

Press [MTH][REAL][IP]

5) FP

Show the Fraction Part of a number

Example:

Level 2:

Press [MTH][REAL][FP]

6) DR

Converts from Degree to radians

Example:

Level 2:

Press [MTH][REAL][DR]

Result:

7) RD

Converts from Radians to Degree

Example:

Level 2:

8) CEIL

Returns the mayor integer closest to the number

Example:

Level 2:

9) FLOOR

Returns the minor integer closest to the number

Example:

Level 2:

Press [MTH][REAL][FLOOR]

10) %

Returns the percentage

Example:

Level 2:

Level 1:

Press [MTH][REAL][%]

1) EGV

The command**EGV ** calculates the **eigenvalues** and the **eigenvectors** of a given matrix.

Example:

Calculates the eigenvalues and the eigenvectors of the matrix

**Resolution:**

1- write the matrix in the HP48/49 as below:

and press [ENTER]

2- type**EGV** and press [ENTER]

It will show:

where the columns of the matrix are the eigenvectors associated

to the eigenvalues shown in the vector below the matrix.

In this Example:,

[1 0 0] is the eigenvector associated to 3

[1 1 0] is the eigenvector associated to 2

[3 3 1] is the eigenvector associated to 1

The command**EGVL** calculate the eigenvector of a given matrix

Example:

Calculates the eigenvalues and the eigenvectors of the matrix

**Resolution:**

1- write the matrix in the HP48/49 as below:

and press [ENTER]

2- type**EGVL** and press [ENTER]

It shows the vector [ 3 2 1]

where**3**, **2** and **1** are the eigenvalues for the matrix.
**Note:**

These Examples was done with**HP49 mode set RPN**

HP48 is also able to calculate eigenvalues and eigenvectors,

but HP48 shows the eigenvector matrix in line,

and eigenvalues as a vector.

For the Examples of**EGV** above it shows something like,

**[ [ 1 1 3 ] [ 0 1 3 ] [ 0 0 1] ]**

**[ 3 2 1]**.

Press [SWAP] to see it like.

**[ 3 2 1]**

**[ [ 1 1 3 ]**

[ 0 1 3 ]

[ 0 0 1 ] ]

Where the eigenvectors are shown as colums of the matrix,

and the eigenvalues as the vector**[ 3 2 1]**.

The command

Example:

Calculates the eigenvalues and the eigenvectors of the matrix

1- write the matrix in the HP48/49 as below:

and press [ENTER]

2- type

It will show:

where the columns of the matrix are the eigenvectors associated

to the eigenvalues shown in the vector below the matrix.

In this Example:,

[1 0 0] is the eigenvector associated to 3

[1 1 0] is the eigenvector associated to 2

[3 3 1] is the eigenvector associated to 1

The command

Example:

Calculates the eigenvalues and the eigenvectors of the matrix

1- write the matrix in the HP48/49 as below:

and press [ENTER]

2- type

It shows the vector [ 3 2 1]

where

These Examples was done with

HP48 is also able to calculate eigenvalues and eigenvectors,

but HP48 shows the eigenvector matrix in line,

and eigenvalues as a vector.

For the Examples of

Press [SWAP] to see it like.

[ 0 1 3 ]

[ 0 0 1 ] ]

Where the eigenvectors are shown as colums of the matrix,

and the eigenvalues as the vector