# Addition, Subtraction & Multiplication | Algebra - Number System

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**Addition, Subtraction & Multiplication | Algebra - Number System**

Add (1101)_{2} and (101)_{2}. Write your answer in base 2.

Answer: 10010

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**Explanation** :

When we add numbers in non-decimal system, add them as we do in decimal system but write them according to the given base system.

**Step 1:** Adding unit’s digits.

Add them as decimal system i.e., 1 + 1 = 2.

Now write 2 in binary system i.e., (2)_{10} = (11)_{2}

Write 0 and carry over 1 as we do with normal addition.

**Step 2:** Adding ten’s digits.

Add them as decimal system i.e., 1 + 0 + 0 = 1.

**Step 3:** Adding hundred’s digits.

Add them as decimal system i.e., 1 + 1 = 2.

Now write 2 in binary system i.e., (2)_{10} = (10)_{2}

Write 0 and carry over 1.

**Step 4:** Adding thousand’s digits.

Add them as decimal system i.e., 1 + 1 = 2.

Now write 2 in binary system i.e., (2)_{10} = (10)_{2}

Hence, 10010.

Workspace:

**Addition, Subtraction & Multiplication | Algebra - Number System**

Add (1034)_{5} and (210)_{5}. Write your answer in base 5.

Answer: 1244

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**Explanation** :

When we add numbers in non-decimal system, add them as we do in decimal system but write them according to the given base system.

**Step 1:** Adding unit’s digits.

**Step 2:** Adding ten’s digits.

**Step 3:** Adding hundred’s digits.

**Step 4:** Adding thousand’s digits.

Hence, 1244.

Workspace:

**Addition, Subtraction & Multiplication | Algebra - Number System**

Add (1357)_{8} and (2672)_{8}. Write your answer in base 8.

Answer: 4251

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**Explanation** :

When we add numbers in non-decimal system, add them as we do in decimal system but write them according to the given base system.

**Step 1:** Adding unit’s digits, 7 + 2 = 9 = (11)_{8}

**Step 2:** Adding ten’s digits, 1 + 5 + 7 = 13 = (15)_{8}

**Step 3:** Adding hundred’s digits, 1 + 3 + 6 = 10 = (12)_{8}

**Step 4:** Adding thousand’s digits, 1 + 1 + 2 = 4 = (4)_{8}

Hence, 4251.

Workspace:

**Addition, Subtraction & Multiplication | Algebra - Number System**

Calculate (1101)_{2} - (101)_{2}. Write your answer in base 2.

Answer: 1000

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**Explanation** :

When we subtract numbers in non-decimal system, subtract them as we do in decimal system but write them according to the given base system.

**Step 1:** Subtracting unit’s digits.

Subtract them as decimal system i.e., 1 - 1 = 0.

**Step 2: **Subtracting ten’s digits.

Subtract them as decimal system i.e., 0 + 0 = 0.

**Step 3:** Subtracting hundred’s digits.

Subtract them as decimal system i.e., 0 - 0 = 0.

**Step 4:** Subtracting thousand’s digits.

Subtract them as decimal system i.e., 1 - 0 = 1.

Hence, 1000.

Workspace:

**Addition, Subtraction & Multiplication | Algebra - Number System**

Calculate (100)_{2} - (11)_{2}. Write your answer in base 2.

Answer: 1

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**Explanation** :

When we subtract numbers in non-decimal system, subtract them as we do in decimal system but write them according to the given base system.

**Step 1**: Subtracting unit’s digits.

When we subtract bigger number from a smaller number, we borrow from the left of the smaller digit.

Since we are dealing in base 2, when we borrow, we subtract 1 from the left digit and add 2 in the smaller digit.

Now subtract as we do in decimal system i.e., 2 + 0 – 1 = 1

**Step 2**: Subtracting ten’s digits.

When we subtract bigger number from a smaller number, we borrow from the left of the smaller digit.

Since we are dealing in base 2, when we borrow, we subtract 1 from the left digit and add 2 in the smaller digit.

Now subtract as we do in decimal system i.e., 2 – 1 + 0 - 1 = 0.

**Step 3:** Subtracting hundred’s digits.

Subtract as we do in decimal system i.e., -1 + 1 – 0 = 0.

Hence, 1.

Workspace:

**Addition, Subtraction & Multiplication | Algebra - Number System**

Calculate (235)_{8} - (66)_{8}. Write your answer in base 8.

Answer: 147

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**Explanation** :

When we subtract numbers in non-decimal system, subtract them as we do in decimal system but write them according to the given base system.

**Step 1:** Subtracting unit’s digits.

When we subtract bigger number from a smaller number, we borrow from the left of the smaller digit.

Since we are dealing in base 8, when we borrow, we subtract 1 from the left digit and add 8 in the smaller digit.

Now subtract as we do in decimal system i.e., 8 + 5 – 6 = 7.

**Step 2:** Subtracting ten’s digits.

When we subtract bigger number from a smaller number, we borrow from the left of the smaller digit.

Since we are dealing in base 8, when we borrow, we subtract 1 from the left digit and add 8 in the smaller digit.

Now subtract as we do in decimal system i.e., 8 – 1 + 3 – 6 = 4.

**Step 3:** Subtracting hundred’s digits.

Now subtract as we do in decimal system i.e., – 1 + 2 = 1.

Hence, 147.

Workspace:

**Addition, Subtraction & Multiplication | Algebra - Number System**

Multiply (22)_{3} and (22)_{3}. Write your answer in base 3.

Answer: 2101

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**Explanation** :

When we multiply numbers in non-decimal system, multiply them as we do in decimal system but write them according to the given base system.

**Step 1**: Multiplying unit’s digits.

Multiply them as decimal system i.e., 2 × 2 = 4.

Now write 4 in base 3 i.e., (4)_{10} = (11)_{3}

Write 1 and carry over 1 as we do with normal multiplication.

**Step 2:** Multiplying unit’s and ten’s digits.

Multiply them as decimal system i.e., 2 × 2 + 1 = 5.

Now write 5 in base 3 i.e., (5)_{10} = (12)_{3}

**Step 3:** Multiplying ten’s and unit’s digits.

Multiply them as decimal system i.e., 2 × 2 = 4.

Now write 4 in base 3 i.e., (4)_{10} = (11)_{3}

**Step 4:** Multiplying ten’s digits.

Multiply them as decimal system i.e., 2 × 2 + 1= 5.

Now write 5 in base 3 i.e., (5)_{10} = (12)_{3}

**Step 5:** Add the two results obtained.

∴ (22)_{3} × (22)_{3} = (2101)_{3}.

Hence, 2101.

Workspace:

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