Addition, Subtraction & Multiplication

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

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​​​​​​​Hence, 10010.

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.

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Hence, 1244.

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

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Hence, 4251.

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.

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Hence, 1000.

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.

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Hence, 1.

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.

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Hence, 147.

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.