Integers

Imagine a world with two types of particles: positives (+) and negatives (-). They interact in specific ways.

• When particles of the same type meet, they join forces.
• When a positive and a negative particle meet, they cancel each other out, leaving nothing. This is called a zero pair.
• Let’s see what happens if 2 positives meet 1 negative. One zero pair is formed, leaving 1 positive.
• To show which type of particle we have, we put a sign in front of the number:
  • The + sign for a group of positives.
  • The - sign for a group of negatives.
• Now, let’s see what happens when 3 positives meet 1 negative. There are 2 positives left.
• Finally, let’s see what happens when 2 positives meet 2 negatives. There are \(0\) particles left.

• • For the subtraction, \(\textcolor{colordef}{(+3)}-\textcolor{colordef}{(+2)}\):we remove \(2\) positives from \(3\) positives, leaving us with \(1\) positive.
  • For the addition, \(\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-2)}\):we again remove \(2\) positives from \(3\) positives.
  • Therefore, these two operations are equivalent:\(\textcolor{colordef}{(+3)}-\textcolor{colordef}{(+2)}=\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-2)}\)This shows that subtracting a positive number is the same as adding its opposite.
• • For the subtraction, \(\textcolor{colorprop}{(-3)}-\textcolor{colorprop}{(-2)}\):we remove \(2\) negatives from \(3\) negatives, leaving us with \(1\) negative.
  • For the addition, \(\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+2)}\):we again remove \(2\) negatives from \(3\) negatives.
  • Therefore, these two operations are equivalent:\(\textcolor{colorprop}{(-3)}-\textcolor{colorprop}{(-2)}=\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+2)}\)This shows that subtracting a negative number is the same as adding its opposite.
• In conclusion, these examples show a fundamental rule in arithmetic: subtracting any number is equivalent to adding the number with its opposite sign.

• To show both positive and negative numbers on a number line, we extend the number line in both directions from zero.
• For each move from left to right by \(1\), the number increases by \(1\): \(0+1=\textcolor{colordef}{+1}\), \(\textcolor{colordef}{+1}+1=\textcolor{colordef}{+2},\dots\)
• For each move from right to left by \(1\), the number decreases by \(1\): \(0-1=\textcolor{colorprop}{-1}\), \(\textcolor{colorprop}{-1}-1=\textcolor{colorprop}{-2},\dots\)

In the set of integers, the order is defined as:$$\dots\lt\textcolor{colorprop}{-3}\lt\textcolor{colorprop}{-2}\lt\textcolor{colorprop}{-1}\lt 0 \lt \textcolor{colordef}{+1}\lt\textcolor{colordef}{+2}\lt\textcolor{colordef}{+3}\lt\dots $$So, as you move along the number line from left to right, the numbers increase.

• As \(\textcolor{colordef}{+3}\) is to the right of \(\textcolor{colorprop}{-5}\), \(\textcolor{colorprop}{-5} \lt \textcolor{colordef}{+3}\). So, when one number is positive and the other is negative, the positive number is greater.
• As \(\textcolor{colorprop}{-2}\) is to the right of \(\textcolor{colorprop}{-4}\), \(\textcolor{colorprop}{-4}\lt \textcolor{colorprop}{-2}\). So, when both numbers are negative, the number closer to zero is greater (the number with the smaller absolute value is greater).
• As \(\textcolor{colordef}{+6}\) is to the right of \(\textcolor{colordef}{+4}\), \(\textcolor{colordef}{+4}\lt \textcolor{colordef}{+6}\). So, when both numbers are positive, the number further from zero is greater (the number with the greater absolute value is greater).