Key Points

An arithmetic expression is a combination of numbers using operations like addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$). Every expression evaluates to a single number, which is its value. For example, the value of the expression $15 - 3$ is $12$.

Terms are the parts of an expression separated by addition ($+$) or subtraction ($-$) signs. Subtraction is treated as adding the inverse, so $a - b$ is the same as $a + (-b)$. For example, in $23 - 2 \times 4 + 16$, the terms are $23$, $-2 \times 4$, and $16$.

To find the value of an expression without brackets, first evaluate each term (performing any multiplication or division within them), and then add the resulting values. For example, in $30 + 5 \times 4$, first calculate $5 \times 4 = 20$, then add $30 + 20$ to get $50$.

Brackets are used to change or clarify the order of operations. The part of the expression inside the brackets must be evaluated first. For example, in $100 - (15 + 56)$, you first evaluate $15 + 56 = 71$, and then calculate $100 - 71 = 29$.

The order of adding terms does not change the sum. This is known as the commutative property of addition. For any numbers $a$ and $b$, the rule is $a + b = b + a$.

When adding three or more terms, the way they are grouped does not affect the final sum. This is the associative property of addition. For any numbers $a$, $b$, and $c$, the rule is $(a + b) + c = a + (b + c)$.

If brackets are preceded by a '+' sign, you can remove them without changing the signs of the terms inside. For example, $28 + (35 - 10)$ is the same as $28 + 35 - 10$.

If brackets are preceded by a '-' sign, you must invert the sign of each term inside the brackets when you remove them. For example, $200 - (40 + 3) = 200 - 40 - 3$, and $500 - (250 - 100) = 500 - 250 + 100$.

Multiplying a number by a sum is the same as multiplying the number by each addend separately and then adding the products. This is the distributive property: $a \times (b + c) = (a \times b) + (a \times c)$.

Multiplying a number by a difference is the same as multiplying the number by each term separately and then subtracting the products. The rule is: $a \times (b - c) = (a \times b) - (a \times c)$.

Expressions can often be compared using reasoning instead of full calculation. For example, to compare $1023 + 125$ and $1022 + 128$, notice the first number decreases by $1$ while the second increases by $3$. The right side is greater by $2$. Thus, $1023 + 125 < 1022 + 128$.