What is an Arithmetic Expression?
An arithmetic expression is just a math sentence that uses:
- Numbers \((1, 2, 3 ...)\) and
- Operations like addition \((+)\), subtraction \((-)\), multiplication \((\times)\) or division \((\div)\).
Let's Try Together!
I'll say a situation, and you tell me the expression!
1. You have \(7\) chocolates and get \(5\) more. What is the expression?
Ans: \(7 + 5\)
2. You share \(24\) pencilas with \(4\) friends. What is the expression?
Ans: \(24 \div 4\)
Let's compare arithmetic expressions!
The signs used to compare two expressions:
- \(=\) if both sides are the same.
- \(>\) if the left side value is bigger.
- \(<\) if the right side value is bigger.
Swapping (Commutative Property)
When you add or (multiply) two or more numbers, you can change their order, and the sum or (product) will stay the same.
Grouping (Associative Property)
When you add or (multiply) three or more numbers, you can group them in any way, and the sum or (product) will still be the same
In addition and multiplication you can swap or group numbers any way you like - the final sum and product remains the same.
Important!
Subtraction is neither commutative nor associative because changing the order or grouping changes the difference.
Division is neither commutative nor associative because changing the order or grouping changes the quotient.
Division is neither commutative nor associative because changing the order or grouping changes the quotient.
Distributing a negative sign:
Golden Rule:
Distributing a negative sign flips the sign of every term inside the parentheses (bracket).
Positive becomes negative, and negative becomes positive.
Distributing a negative sign flips the sign of every term inside the parentheses (bracket).
Positive becomes negative, and negative becomes positive.
\(- (a + b) = - a - b\)
\(- (-a + b) = a - b\)
\(- (a - b) = -a + b\)
\(- (-a - b) = a + b\)
Distributive property of integers
Let's say we have the expression \(2 \times (40 + 10)\).
What We normally do:
(i) First add inside the bracket: \(40 + 10 = 50\)
(ii) Then, multiply: \(2 \times 50 = 100\)
Now, what if we don't use the bracket?
Do this instead:
\(2 \times 40 + 2 \times 10 = 80 + 20 = 100\)
The answer is same.
(i) First, do the subtraction inside the bracket: \(50 - 20 = 30\)
(ii) Then, multiply: \(3 \times 30 = 90\)
Now, split it: \(3 \times 50 - 3 \times 20 = 150 - 60 = 90\)
The answer is same.
Sometimes, multiplying big numbers directly can feel hard. But what if we change one of the numbers to make it easier? We break it into simpler parts and then solve!
What We normally do:
(i) First add inside the bracket: \(40 + 10 = 50\)
(ii) Then, multiply: \(2 \times 50 = 100\)
Now, what if we don't use the bracket?
Do this instead:
\(2 \times 40 + 2 \times 10 = 80 + 20 = 100\)
The answer is same.
Let's try with subtraction:
Take the expression \(3 \times (50 - 20)\) (i) First, do the subtraction inside the bracket: \(50 - 20 = 30\)
(ii) Then, multiply: \(3 \times 30 = 90\)
Now, split it: \(3 \times 50 - 3 \times 20 = 150 - 60 = 90\)
The answer is same.
If there's a number outside the bracket, you can multiply it with each number inside, and then add or subtract. This is called distributive property.
We are Tinkering (Changing) the Number!
\((a + b) \times c = a \times c + b \times c\)
\((a - b) \times c = a \times c - b \times c\)
\((a + b) \times c = a \times c + b \times c\)
\((a - b) \times c = a \times c - b \times c\)