Integer foundations

Integers extend whole numbers so we can model values above, below, and exactly at a reference point. They appear in temperature, sea level, game scores, electricity, and money balances. Without integers, many real situations would be awkward to calculate.

A short history of the number line idea

The modern number line grew over time. Early printed works arranged numbers in order, and later mathematicians described arithmetic as movement along a line. By the late 1600s, this “walk forward / backward” idea was used to explain addition and subtraction in a geometric way.

The key shift was not just drawing numbers in order, but treating position on a line as a model for operations. That model is now foundational in school mathematics.

Drawing and reading a number line

A number line is a straight line with equally spaced marks. Numbers increase to the right and decrease to the left. Zero is the center reference point for positive and negative values.

• Right of 0: positive integers (\(+1, +2, +3, \dots\)).
• Left of 0: negative integers (\(-1, -2, -3, \dots\)).
• At 0: neither positive nor negative.

In coordinate graphs, both axes are number lines: the horizontal \(x\)-axis and vertical \(y\)-axis.

Comparing integers with position

Position gives order immediately:

• If \(a\) is to the right of \(b\), then \(a > b\).
• If \(a\) is to the left of \(b\), then \(a < b\).

Example: \(-2 > -5\) because \(-2\) is to the right of \(-5\), even though both are negative.

Absolute value and opposites

Absolute value means distance from zero, so it is never negative: \(|-7|=7\), \(|7|=7\). Opposite numbers are equally far from zero in opposite directions, such as \(+5\) and \(-5\).

Distance between two integers can be read from the line: \[ \text{distance between } a \text{ and } b = |a-b|. \] For example, the distance from \(-4\) to \(3\) is \(|-4-3|=7\).

Operations as movement and grouping

Addition and subtraction

Think of arithmetic as moves on the line:

• \(a+b\): start at \(a\), move \(b\) units (right if \(b>0\), left if \(b<0\)).
• \(a-b = a+(-b)\): subtracting means adding the opposite.

Examples: \[ -3+8=5,\qquad 4-9=-5,\qquad 6-(-4)=10. \]

Multiplication

Multiplication can be seen as repeated equal jumps. For instance, \(5\times3\) means three jumps of length 5 to the right, ending at 15.

Sign rules come from direction:

• \((+)\times(+)=+\)
• \((+)\times(-)=-\)
• \((-)\times(+)=-\)
• \((-)\times(-)=+\)

Division

Division asks how many equal lengths fit into another length. For example, \(6\div2=3\) because three jumps of 2 reach 6. Integer sign rules mirror multiplication: \[ (+)\div(+)=+,\quad (+)\div(-)=-,\quad (-)\div(+)=-,\quad (-)\div(-)=+. \]

Zero principle with tiles

A positive tile and a negative tile form a zero pair. Add or remove zero pairs to simplify without changing value.

Checkpoint: evaluate \(-3 + 8\)

\(-3 + 8 = 5\). Start at \(-3\), move 8 units right.

Checkpoint: evaluate \(4 - 9\)

\(4 - 9 = -5\). Equivalent to \(4 + (-9)\).

Checkpoint: explain why \(2-(-2)=4\)

Subtracting \(-2\) means adding its opposite \(+2\), so \(2-(-2)=2+2=4\).

Why zero and negatives matter

Zero and negative numbers make arithmetic coherent. Before negatives were accepted, people often restricted results to non-negative values, which forced case-by-case rules and extra wording.

With integers, one set of laws works smoothly:

• Subtraction is always possible: \(a-b=a+(-b)\).
• Opposites are explicit: every \(a\) has \(-a\), and \(a+(-a)=0\).
• Distributive structure stays consistent across signs: \(k(a+b)=ka+kb\), even when values are negative.

This consistency is why integer arithmetic is not just “extra symbols”; it is the foundation for algebra, equations, and later topics like vectors and coordinate geometry.