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Using Julia as a calculator¶

Quick background¶

Read about this material here: Julia as a calculator.

For the impatient, this project covers the use of Julia to replace what a calculator can do.

Using Jupyter¶

This workspace is a Jupyter notebook. For a bit of orientation:

• Commands are typed into cells. There can be one or more commands in a cell. Use a semicolon or newline to separate commands.
• Cells are executed in different ways: using “shift-enter,” using the “Run” button in the tool bar, or using the “Cell” menu in the menu bar.
• Output from execution appears below the cell. Output shows the last command executed
• After a cell is executed, a running tally of what was executed when is displayed next to the In [...] and Out[...] labels.
• The actually running of a cell is executed in a kernel which may be on a different machine.
• Some cells – such as this one – are marked Markdown in the toolbar, this is for formatted text. You will type commands into cells designated as “Code.”

The project notebooks¶

The project notebooks, like this one, offer a brief introduction into a topic and then allow you to continue within the notebook. There is also a blank notebook to use without introductory text.

It is suggested that you use the “Run” menu to “Run all cells” when these notebook are loaded.

The common operations on numbers: addition, subtraction, multiplication, division, and powers.¶

For the most part there is no surprise, once you learn the notations: +, -, *, /, and ^. (Though you may find that copying and pasting minus signs will often cause an error, as only something that looks like a minus sign is pasted in.)

Using IJulia, one types the following into a cell and then presses the run button (or shift-enter):

The answer follows below the cell.

Here is how one does a slightly more complicated computation:

As with your calculator, it is very important to use parentheses, as appropriate, to circumvent the usual order of operations.

One aspect that can be surprising is multiplication by a literal number does not require a * to be typed (e.g. 2x or 2*x). However, when no * is given, the order of operation is different than may be expected.

The use of the basic families of function: trigonometric, exponential, logarithmic.¶

On a calculator, there are buttons used to compute various functions. In Julia, there are many pre-defined functions that serve a similar role (In the next project you will see how to define your own).

Functions in Julia have names and are called using parentheses to enclose their argument(s), as with:

(With IJulia, when a cell is executed only the last command computed is displayed, the above shows that using a comma to separate commands on the same line can be used to get two or more commands to be displayed.)

Most basic functions in Julia have easy to guess names, though you will need to learn some differences, such as log is for $\ln$ and asin for $\sin^{-1}$. Some function names encountered in this class include: sqrt, cbrt, sin, cos, tan, asin, acos, atan,exp, and log.

The trigonometric functions use radians. For degree measure the conversion factor $\pi/180$ must be judiciously employed.

The function $f(x) = e^x$ is implemented with function exp(x) and not using e^x. In base Julia e is not defined.

The use of memory registers to remember intermediate values.¶

Rather than have numbered memory registers like a calculator, it is easy to assign a name to a value. For example,

Names can be reassigned and repurposed. However, a function name can not be repurposed as a variable name and a variable name can not be repurposed as a function name.

You will get an ERROR message if this is tried. (A suggestion is to use longer descriptive names or follow math conventions: use x, y, z, … for variables; f and g for functions; and a,b, … for parameters, though even then values like h may play a dual role)

For assigning more than one value at once, commas can be used as with:

The equals sign, =, in computer languages is not the same, as it is in mathematics. In math it typically sets up an equation. On the computer it is reserved for assignment. As such only a limited number of expressions can occur on the left-hand side — things being assigned.

Julia, like math, has different number types¶

Unlike a calculator, but just like math, Julia has different types of numbers: integers, rational numbers, real numbers, and complex numbers. For the most part the distinction isn’t much to worry about, but there are times where one must, such as overflow with integers. (One can only take the factorial of 20 with 64-bit integers, whereas on most calculators a factorial of 69 can be taken, but not 70.) Julia automatically assigns a type when it parses a value: a 1 will be an integer, a 1.0 an floating point number. Rational numbers are made by using two division symbols, 1//2.

For many operations the type will be conserved, such as adding two integers. For some operations, the type will be converted, such as dividing two integer values. Mathematically, we know we can divide some integers and still get an integer, but Julia usually opts for the same output for its functions (and division is also a function) based on the type of the input, not the specific values of the input.

Numbers¶

Scientific notation represents real numbers as $\pm a \cdot 10^b$, where $b$ is an integer, and $a$ may be a real number in the range $[1.0, 10)$. In Julia such numbers are represented with an e to replace the 10, as with 1.2e3 which would be $1.2 \cdot 10^3$ (better known as 1,230) or 3.2e-1, which would be $3.2 \cdot 10^{-1}$ (which is equal to $0.32$).

Take note that this $e$ is not the special base of the natural logarithm, but is just notation indicating a power of $10$. To use this notation, you must have a number immediately before the “e”, not a space or an asterisk. That is, these are not correct: 1.2*e3 or 12 e 3.