NCERT Solutions: Class 6 Maths Chapter 10 - The Other Side of Zero

Going up is represented by a positive movement; going down is represented by a negative movement.

The Welcome Hall is Floor $0$. Floors above it are positive and floors below it are negative.

Starting floor plus movement gives target floor: $(+2)+(-3)=-1$.

Add the signed movements directly, treating upward movement as positive and downward movement as negative.

For each starting floor, choose the movement that makes starting floor + movement equal $-5$.

Combine all upward and downward movements: c. $4-3-2=-1$; d. $-1+2-3=-2$.

The inverse of a number is the number with the opposite sign. The inverse of $0$ is $0$.

The lowest floor is the smallest integer. Among $+2,+5,-3,-1$, the smallest is $-3$.

On the number line, numbers farther right are greater. Positive numbers are greater than negative numbers, and $0$ is greater than negative numbers but less than positive numbers.

For negative numbers, the number closer to zero is greater. Any positive number is greater than any negative number.

Think of subtraction as the missing movement: $5+10=15$, $10+90=100$, and $34+40=74$.

Subtract the starting floor from the target floor to find the movement needed.

Find the signed movement that makes each equation true; for subtraction, compute target minus starting value.

Moving from a negative starting value to a target is the same as adding the corresponding positive distance. For example, $+2000-(-200)=+2000+(+200)=+2200$.

Subtracting a negative number is the same as adding the corresponding positive number. Then combine signs normally.

Positive numbers are marked to the right of $0$; negative numbers are marked to the left of $0$.

List the negative numbers that were marked on the number line.

On the number line, $2$ is to the right of $-3$, so $2>-3$. Also, $-2$ is to the left of $3$, so $-2<3$.

Use signed addition/subtraction: $-5+0=-5$, $7-7=0$, $-10+20=10$, $10-20=-10$, $7-(-7)=14$, and $-8-(-10)=2$.

Combine positive and negative tokens, cancelling zero pairs where possible.

Subtract by taking away the requested tokens; add zero pairs when needed. The arithmetic matches ordinary integer subtraction.

Compute each difference using integer subtraction rules.

Start with $3$ negative tokens. To take away $5$ positive tokens, add $5$ zero pairs. After removing $5$ positives, $8$ negatives remain.

Use zero pairs when needed, or compute directly: subtracting a positive moves left; subtracting a negative moves right.

Starting balance $Rs\ 100$ plus credit $Rs\ 60$ gives $100+60=160$.

After paying $Rs\ 30$, the balance is $160-30=130$.

After a debit of $Rs\ 150$, the balance is $130-150=-20$.

A credit of $Rs\ 200$ after balance $-Rs\ 20$ gives $-20+200=180$.

Credits total $30+40+50=120$. Debits total $40+50+60=150$. Balance $=120-150=-30$.

Debits total $1+2+4+8+16+32+64+128=255$. Then $256-255=1$.

Negative balances can create extra charges, so they are usually avoided. They may be worthwhile only when the benefit of the purchase or emergency payment is greater than the temporary cost.

From the labelled heights, A has the greatest positive height and D has the lowest negative height.

Order the approximate heights from greatest to least and then reverse for increasing order.

Mount Everest is the highest point above sea level on Earth.

Depths below sea level are represented by negative numbers. Challenger Deep is about $10994\text{ m}$ below sea level.

Temperature usually decreases with altitude, and mountainous northern regions receive colder winter conditions, so temperatures can fall below $0^\circ\text{C}$.

The warmest temperature is most likely afternoon, the next warmest late morning, and the coldest temperatures occur at night or early morning.

For the second grid, each border row or column sum is $-3$, so the border sum is $-3$.

Repeat the grid procedure with different choices and compare the resulting sums to detect the pattern.

The entries are arranged so that different allowed choices produce controlled sums. Similar grids can be made by arranging numbers in arithmetic patterns.

List only the integers strictly between each pair, from smaller to larger.

$-5+7-10=-8$. Many other triples are possible.

Form all pairwise sums from the set $\{-5,-3,-1,2,4,6\}$. The listed integers between $-10$ and $12$ do not occur among those sums.

Evaluate each signed expression in order: $8-13=-5$, $-8-13=-21$, $-13-(-8)=-5$, $-13+(-8)=-21$, $8+(-13)=-5$, $-8-(-13)=5$, $13-8=5$, and $13-(-8)=21$.

a. $2026-150=1876$. b. Going back $2200$ years from $2026\text{ CE}$ crosses from $1\text{ CE}$ directly to $1\text{ BCE}$ with no year $0$, giving $175\text{ BCE}$. c. $680\text{ BCE}$ plus $320$ years is $360\text{ BCE}$.

a. Add $6$ each time. b. The pattern alternates one less on the first subsequence and one more on the second. c. The differences are $-8,-7,-6,-5,-4,-3,-2,-1,0$.

Compute $-2-9-18-1=-30$, which is exactly $-30$.

Subtracting a negative adds a positive, so part a is positive. Adding or subtracting mixed signs depends on magnitudes. Adding two negatives or subtracting a positive from a negative gives a negative.

Subtracting a smaller positive from a larger positive moves right of zero; subtracting a larger positive from a smaller positive moves left of zero; subtracting a negative is moving upward/right; subtracting a number from itself returns to zero; subtracting zero leaves the position unchanged.

Each example matches one of Brahmagupta's subtraction rules stated in the text.