Prerequisites
Arithmetic | 1 of 17
Prerequisites: None. Enables: Addition, Integers and Signed Arithmetic.
Arithmetic studies numbers and the basic operations used to combine, compare, and transform them.
The main arithmetic operations are:
Addition
Subtraction
Multiplication
Division
Number sets
| Number set | Description | Examples |
|---|---|---|
| Natural numbers | Counting numbers; some texts also include $0$, so check the convention | $1,\ 2,\ 3,\ 4$ |
| Whole numbers | Natural numbers and zero | $0,\ 1,\ 2,\ 3$ |
| Integers | Positive numbers, negative numbers, and zero | $-3,\ -2,\ -1,\ 0,\ 1$ |
| Rational numbers | Numbers expressible as $p/q$ for integers $p,q$ with $q\ne0$ | $\frac{1}{2},\ -\frac{5}{3},\ 0.75$ |
| Irrational numbers | Real numbers that cannot be expressed as such a ratio | $\sqrt{2},\ \pi$ |
| Real numbers | Rational and irrational numbers together | $-4,\ 0,\ \frac{2}{3},\ \sqrt{5}$ |
Arithmetic usually begins with whole numbers, then extends to integers, fractions, decimals, and real numbers. A decimal is rational exactly when its decimal expansion terminates or repeats; this is a consequence of the ratio definition, not the definition itself.
Place value
In base ten, each digit's value depends on its position.
For the number:
$$
4{,}382.65
$$
the place values are:
| Place | Digit | Digit's value |
|---|---|---|
| Thousands | 4 | 4,000 |
| Hundreds | 3 | 300 |
| Tens | 8 | 80 |
| Ones | 2 | 2 |
| Tenths | 6 | 0.6 |
| Hundredths | 5 | 0.05 |
Expanded form:
$$
4{,}382.65 = 4000 + 300 + 80 + 2 + 0.6 + 0.05
$$
Comparing numbers
Use the symbols:
| Symbol | Meaning |
|---|---|
| $=$ | equal to |
| $\ne$ | not equal to |
| $<$ | less than |
| $>$ | greater than |
| $\le$ | less than or equal to |
| $\ge$ | greater than or equal to |
Examples:
$$
7 > 3
$$
$$
-5 < -2
$$
$$
0.25 = \frac{1}{4}
$$
Absolute value
The absolute value of a number is its distance from zero.
$$
|a| \ge 0
$$
Examples:
$$
|6| = 6
$$
$$
|-6| = 6
$$
$$
|0| = 0
$$