Integers

The integers are the whole numbers together with their negatives and zero:

Integers do not include fractions or decimals. A positive integer is greater than $0$, a negative integer is less than $0$, and $0$ itself is neither positive nor negative.

Opposites sit the same distance from zero.

The absolute value of a number is its distance from $0$ on the number line. Distance is never negative, so absolute value is always $0$ or positive. We write it with bars: $|{-6}| = 6$ and $|6| = 6$.

Opposites sit the same distance from zero.

On the number line, the number farther to the right is greater. So any positive number is greater than any negative number, and for two negatives, the one closer to $0$ is greater. That means $-2 > -7$, because $-2$ is to the right of $-7$.

Same signs: add the absolute values and keep the common sign. Different signs: subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value.

To subtract an integer, add its opposite. Change the subtraction to addition and flip the sign of the number being subtracted:

Then follow the rules for adding integers.

Opposites sit the same distance from zero.

Multiply the absolute values, then decide the sign:

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• positive $\times$ positive $=$ positive
• negative $\times$ negative $=$ positive
• positive $\times$ negative $=$ negative (either order)

Same signs give a positive; different signs give a negative.

Division follows the exact same sign rules as multiplication. Divide the absolute values, then apply: same signs $\rightarrow$ positive, different signs $\rightarrow$ negative.

Follow the usual order: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right). The only new thing is keeping careful track of signs at every step.

Real situations often use negatives: temperatures below zero, money owed (debt), and elevation below sea level. Rising, gaining, and depositing are positive; falling, losing, and spending are negative.

The rule “a negative times a negative is positive” is not arbitrary---it is forced by the distributive property, $a(b+c) = ab + ac$. Watch:

But distributing the same expression gives

Both equal $0$, so $-1 + (-1)(-1) = 0$, which means $(-1)(-1)$ must be $+1$. Any other value would break arithmetic. Two negatives must make a positive.

Opposites sit the same distance from zero.

A set is closed under an operation if combining any two members always lands you back inside the set. The integers are closed under addition, subtraction, and multiplication: adding, subtracting, or multiplying two integers always yields another integer. They are not closed under division, since $3 \div 2 = 1.5$ is not an integer. Other key properties for all integers $a,b,c$:

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• Commutative (add, multiply): $a+b = b+a$ and $ab = ba$.
• Associative (add, multiply): $(a+b)+c = a+(b+c)$.
• Distributive: $a(b+c) = ab + ac$.
• Identities: $a + 0 = a$ and $a \times 1 = a$.
• Additive inverse: every integer $a$ has an opposite $-a$ with $a + (-a) = 0$.

Subtraction is not commutative or associative: $5 - 3 \neq 3 - 5$.

When we divide, the remainder is what is left over. Mathematicians agree that the remainder should always be non-negative, even when the number is negative. So we find $a \bmod n$ by adding or subtracting copies of $n$ until we land in the range $0$ to $n-1$.

For example, $-7 \bmod 5$: add $5$ twice, $-7 + 5 + 5 = 3$, so $-7 \equiv 3 \pmod 5$. Check: $-7 = (-2)(5) + 3$, and the remainder $3$ sits in $\{0,1,2,3,4\}$. This is exactly how a clock works---“$3$ hours before $2$ o'clock” wraps around to $11$.

Because $|x|$ measures distance from $0$, an equation like $|x| = 5$ has two answers: $x = 5$ or $x = -5$ (both are $5$ units away). More generally:

Inequalities split into two shapes:

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• $|x| < k$ means $x$ is within $k$ of zero: $-k < x < k$ (a single band).
• $|x| > k$ means $x$ is farther than $k$ from zero: $x < -k$ or $x > k$ (two pieces).

And $|x| = -3$ has no solution: distance can never be negative.

The sum of the integers from $1$ to $n$ has a famous closed form (Gauss's trick: pair the first with the last):

For example, $1 + 2 + \cdots + 100 = \dfrac{100 \cdot 101}{2} = 5050$. This also reveals a neat fact: the sum of any run of consecutive integers that is symmetric about $0$, such as $-4 + (-3) + \cdots + 3 + 4$, is $0$, because every term cancels its opposite.