1.1 Integers

Integers extend without end in both directions. On a number line, positive integers lie to the right of zero and negative integers lie to the left. The sign tells which side of zero the number is on; zero is neither positive nor negative.

Zero can hold an empty place. In $406\,072$, the zero in the ten-thousands place prevents the $6$ from being read as sixty thousand, and the zero in the hundreds place keeps $72$ in the final two places.

To write a number from words, place each stated value in its column and insert zeros where a place is missing. For example, six thousand and seventy-nine is $6079$, not $679$.

A change is found by final value minus initial value. If a temperature rises from $-4^\circ$C to $3^\circ$C, the change is $3-(-4)=+7^\circ$C. The positive answer records a rise.

A difference is a non-negative distance between values. The difference between $-6^\circ$C and $5^\circ$C is $11^\circ$C because the interval crosses zero: $6+5=11$.

Among negative integers, the number closer to zero is greater. For example, $-3>-8$ because $-3$ lies to the right of $-8$.

To order several integers from smallest to largest: place the most negative values first, then zero, then positive values. Within the negatives, larger absolute value means smaller number.

For $-7, 3, -5, -9, 0, 1$, the ascending order is $-9,-7,-5,0,1,3$. The reverse list gives descending order.

First identify the operation, then handle the signs, calculate the unsigned values, and check whether the answer's sign is sensible. Use the inverse operation to check: $-48\div6=-8$ checks $6\times(-8)=-48$.

Keep units in contextual calculations. If one tunnel is $15\,516$ m long and another is $8\,820$ m long, the difference is $15\,516-8\,820=6\,696$ m.

For $6^2+2^3\times5$, calculate powers first: $36+8\times5$. Then multiply: $36+40$. Finally add, giving $76$.

Brackets can make an intended equality true. In $25+3\times(7-2)$, the bracket gives $25+3\times5=40$. Without it, multiplication occurs before subtraction and the value is different.

$2$ is the only even prime. Every prime greater than $2$ is odd, but not every odd number is prime: $9$ is odd and has factors $1,3,9$.

Factor and multiple statements reverse: if $6$ is a factor of $42$, then $42$ is a multiple of $6$.

A proposed common factor must divide every given number exactly. A proposed common multiple must be divisible by every given number. For example, $6$ divides both $12$ and $18$, while $36$ is divisible by both.