Context: Let us consider the expression again. What happens when we change the order and add and first, and then add this sum to ? Will we get the same sum as before? We see that adding the terms of the expression in any order gives the same sum of . Q. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Understand the Question Addition of integers follows both the commutative property () and the associative property (). This means the terms of an expression can be rearranged, grouped, or added in any order without changing the final sum. We will verify this property by reordering the terms in the given expression, testing a new 3-term expression, and testing an expression with 4 terms. Step 1 · Verify with the Given Expression Given expression: Adding the negative numbers first Adding this result to The sum is , which is the same as the original sum. Step 2 · Test with Another 3-Term Expression Consider the expression . Order 1: Left to right Order 2: Adding and first Both orders produce the same sum of . Step 3 · Test with an Expression Having More Than 3 Terms Consider the 4-term expression . Order 1: Left to right Order 2: Grouping positive and negative terms Both orders yield the same result of . Answer > Yes, adding the terms of an expression in any order gives the same value. Common Mistakes Sign Errors with Negative Numbers: Forgetting that adding two negative numbers yields a negative number with their magnitudes added, e.g., , not . Sign of the Difference: When adding numbers with opposite signs, the sign of the larger absolute value is taken, e.g., .

Arithmetic Expressions | A

Question 3

Context: Let us consider the expression $(-7) + 10 + (-11)$ again. What happens when we change the order and add $-7$ and $-11$ first, and then add this sum to $10$ ? Will we get the same sum as before? We see that adding the terms of the expression $(-7) + 10 + (-11)$ in any order gives the same sum of $-8$ .

Q. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Solution

Understand the Question

Addition of integers follows both the commutative property ( $a + b = b + a$ ) and the associative property ( $(a + b) + c = a + (b + c)$ ).
This means the terms of an expression can be rearranged, grouped, or added in any order without changing the final sum.
We will verify this property by reordering the terms in the given expression, testing a new 3-term expression, and testing an expression with 4 terms.

Step 1 · Verify with the Given Expression

Given expression: $(-7) + 10 + (-11)$

Adding the negative numbers first

\begin{aligned} (-7) + (-11) &= -(7 + 11) \\ &= -18 \end{aligned}

Adding this result to $10$

\begin{aligned} 10 + (-18) &= 10 - 18 \\ &= -(18 - 10) \\ &= -8 \end{aligned}

The sum is $-8$ , which is the same as the original sum.

Step 2 · Test with Another 3-Term Expression

Consider the expression $5 + (-3) + 8$ .

Order 1: Left to right

\begin{aligned} 5 + (-3) &= 5 - 3 \\ &= 2 \end{aligned}

$2 + 8 = 10$

Order 2: Adding $5$ and $8$ first $5 + 8 = 13$

\begin{aligned} 13 + (-3) &= 13 - 3 \\ &= 10 \end{aligned}

Both orders produce the same sum of $10$ .

Step 3 · Test with an Expression Having More Than 3 Terms

Consider the 4-term expression $2 + (-4) + 6 + (-1)$ .

Order 1: Left to right

\begin{aligned} 2 + (-4) &= 2 - 4 \\ &= -2 \end{aligned}

$-2 + 6 = 4$

\begin{aligned} 4 + (-1) &= 4 - 1 \\ &= 3 \end{aligned}

Order 2: Grouping positive and negative terms $2 + 6 = 8$

\begin{aligned} (-4) + (-1) &= -(4 + 1) \\ &= -5 \end{aligned}

\begin{aligned} 8 + (-5) &= 8 - 5 \\ &= 3 \end{aligned}

Both orders yield the same result of $3$ .

Answer

Yes, adding the terms of an expression in any order gives the same value.

Common Mistakes

Sign Errors with Negative Numbers: Forgetting that adding two negative numbers yields a negative number with their magnitudes added, e.g., $(-7) + (-11) = -18$ , not $-4$ .
Sign of the Difference: When adding numbers with opposite signs, the sign of the larger absolute value is taken, e.g., $10 + (-18) = -(18 - 10) = -8$ .

Question 3

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