Arithmetic Expressions

1. What is an arithmetic expression?

An arithmetic expression is a combination of numbers connected by the operations $+,\ -,\ \times,\ \div$. Examples: $7+3$, $12-5\times2$, $(4+6)\div2$. Each valid expression stands for a single number — its value. Evaluating $7+3$ gives the value $10$.

A string like $7+\times3$ is not a valid expression because two operation signs sit together with nothing in between. An expression must "make sense" when read left to right.

2. Equalities and inequalities of expressions

When two expressions have the same value, we join them with an equals sign $(=)$ to form an equality. For example, $4+5=10-1$ because both sides equal $9$. If the values differ, we use $>$ or $<$.

To compare without full computation, you can sometimes reason: $4\times18$ vs $4\times20$ → same first factor, larger second factor wins, so $4\times18 < 4\times20$.

3. Terms of an expression (the "+ / -" view)

An expression built with $+$ and $-$ can be read as a sum of terms. The sign in front of a number belongs to that term.

Because addition is commutative ($a+b=b+a$) and associative ($(a+b)+c=a+(b+c)$), we may add the terms in any order and grouping. This is why $13-5+8$ can be computed as $(13+8)-5=21-5=16$, which is often faster.

4. Brackets and grouping

Brackets show which part of an expression should be treated as a single number and evaluated first. Compare:

• $10-(4+3)=10-7=3$
• $(10-4)+3=6+3=9$

The bracket completely changes the meaning, so brackets are not optional decoration — they carry information about grouping.

5. Order of operations (BODMAS)

When several operations appear without enough brackets, we follow an agreed order so that everyone gets the same value:

6. The distributive law (the "× over +" view)

Multiplication distributes over addition and subtraction. This is one of the most useful laws in all of arithmetic:

Geometrically, a rectangle of width $a$ and length $(b+c)$ has area $a(b+c)$, which equals the two smaller rectangles $ab$ and $ac$ put together.

The distributive law also explains the column method of multiplication you already use. When you compute $23\times14$, you are really doing $23\times(10+4)=23\times10+23\times4=230+92=322$. So the "carry and add" routine is the distributive law in disguise — every digit of one number is multiplied by the whole of the other and the partial products are added.

7. Properties of operations — a summary

Caution: subtraction and division are not commutative or associative: $7-3\neq3-7$ and $(12\div6)\div2\neq12\div(6\div2)$.

8. Inserting brackets to make an equality true

A classic puzzle: place brackets so a given equation becomes true.

9. Translating words into expressions

Reading word problems and writing the matching expression is a core skill.

10. Swapping and changing expressions (number sense)

Ganita Prakash encourages you to predict how the value of an expression changes when you tweak it, without recomputing everything. This trains real number sense.

• Adding to a term: increasing one term of a sum increases the whole sum by the same amount. If $40+27=67$, then $40+30=70$ (we added $3$ more).
• Changing a factor: in a product, multiplying one factor by $2$ doubles the value. Since $12\times5=60$, we get $24\times5=120$.
• Subtracting more: in $a-b$, increasing $b$ decreases the value. As $50-10=40$ becomes $50-15=35$.
• Compensation trick: $49+36 = (50-1)+36 = 86-1 = 85$ — round one term up, then adjust.

This idea of "how the value moves when a part changes" is exactly what prepares you for algebra in the next chapters, where the parts become letter-numbers.

11. Brackets within brackets

Sometimes an expression has brackets inside brackets. Always work from the innermost bracket outward.

Different bracket shapes — round $(\ )$, square $[\ ]$, curly $\{\ \}$ — are just nesting markers; they all mean "do me first".

12. Common mistakes to avoid

• Working strictly left to right and ignoring that $\times,\div$ come before $+,-$.
• Forgetting that the sign in front of a number is part of that term.
• Treating subtraction/division as commutative — they are not.
• Dropping brackets when translating "doubled the sum" type phrases.
• Mis-applying the distributive law, e.g. writing $a\times(b\times c)=ab\times ac$ (wrong — that is only for $+$).

13. Quick revision checklist

• Expression → one value; equality $=$ same value on both sides.
• Order: Brackets → powers → $\times\div$ (left to right) → $+\,-$ (left to right).
• $+$ and $\times$ are commutative & associative; $-$ and $\div$ are not.
• Distributive: $a(b+c)=ab+ac$ — use both directions.
• Brackets change meaning; read terms with their signs.