Mada za sehemu hiiUse ordering skills to solve puzzles in everyday lifeMada 4
- Calculate the square of a number
- Calculate the square root of a number
- Explain the concept of integers
- Perform operations in integers
Integers and Operations with Them
Integers are whole numbers that can be positive, negative, or zero. Positive integers are numbers greater than zero (like +5), negative integers are numbers less than zero (like -3), and zero is neither positive nor negative. In everyday life, we use positive integers to show profits, gains, or things we have. We use negative integers to show debts, losses, or things we owe.
When adding integers, we combine positive and negative numbers. A helpful way to think about this is using the zero pair idea — one positive and one negative cancel each other out to equal zero.
Adding Two Positive Integers
When we add two positive integers, the answer is always positive.
Example: +8 + +5 = +13
If you have 8 Tanzania shillings and you get 5 more shillings, you now have 13 shillings.
Adding Two Negative Integers
When we add two negative integers, the answer is always negative.
Example: -4 + -6 = -10
If you owe 4 thousand shillings and you borrow 6 thousand shillings more, you now owe 10 thousand shillings.
Adding Positive and Negative Integers
When adding a positive and a negative integer, look at their sizes (absolute values). The larger number determines the sign of the answer.
Example: +7 + -4 = +3
Think of it as having a profit of 7 items and a debt of 4 items. Using 4 of the profit items to pay the debt leaves a profit of 3 items.
Example: +4 + -9 = -5
A profit of 4 items cannot pay a debt of 9 items. After using the 4 profit items to reduce the debt, you still owe 5 items, so the answer is -5.
To subtract integers, we use the idea of opposites. The opposite of addition is subtraction, and the opposite of a positive number is a negative number.
Rule: Change the subtraction sign to addition, then change the sign of the number that follows it.
Worked Examples
Example 1: +12 - +5
= +12 + -5 = +7
Example 2: +6 - +9
= +6 + -9 = -3
This means if you have a profit of 6 shillings but need to pay a debt of 9 shillings, you will still owe 3 shillings.
Example 3: -8 - +4
= -8 + -4 = -12
If you already owe 8 thousand shillings and you borrow 4 thousand shillings more, you now owe 12 thousand shillings.
When multiplying integers, we follow these rules:
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
Worked Examples
Example 1: (+3) × (+4) = +12
Three groups of four items give twelve items.
Example 2: (-3) × (-4) = +12
The negative sign appears twice, so the answer is positive.
Example 3: (-3) × (+4) = -12
Three groups of negative four give negative twelve.
Example 4: (+5) × (-3) = -15
Five groups of negative three give negative fifteen.
Division follows the same sign rules as multiplication:
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Negative ÷ Positive = Negative
Worked Examples
Example 1: +36 ÷ +6 = +6
Example 2: -36 ÷ -6 = +6
Example 3: +36 ÷ -6 = -6
Example 4: -36 ÷ +6 = -6
| Operation | Sign of Numbers | Result Sign |
|---|---|---|
| Addition | Same signs | Same as both |
| Addition | Different signs | Sign of larger absolute value |
| Subtraction | Change to addition, then follow addition rules | |
| Multiplication | Same signs | Positive |
| Multiplication | Different signs | Negative |
| Division | Same signs | Positive |
| Division | Different signs | Negative |
In Tanzania, integers are used when tracking money. If Mama Mwajuma sells fruits at the market and makes a profit of +15,000 shillings on one day but has a loss of -8,000 shillings on another day, her total profit is +7,000 shillings. Similarly, if she owes the supplier -20,000 shillings and then borrows another -5,000 shillings, her total debt becomes -25,000 shillings. Understanding integers helps her know exactly how much money she has made or still owes.
Swali
What is the result of ?
Ingia ili kuwasilisha jibu lako na lihesabiwe katika umahiri wako.
Ingia ili kufanya mazoeziMwalimu
Umekwama? Niulize chochote kuhusu mada hii.
Ingia ili kumuuliza Mwalimu wa AI wa Sonza kuhusu swali hili.
Ingia ili kuuliza