Step-by-step explanation:
You want a tutorial and a rationale for writing numbers and doing arithmetic in bases other than 10.
A. Rewrite a number
For integers, an easy way to write a number in a different base is to repeatedly divide by the desired base, and list the remainders right to left.
Example 1: convert 67 to base 3:
- 67÷3 = 22 r 1
- 22÷3 = 7 r 1
- 7÷3 = 2 r 1
- 2÷3 = 0 r 2
67₁₀ = 2111₃
Example 2: convert 44 to base 3:
- 44÷3 = 14 r 2
- 14÷3 = 4 r 2
- 4÷3 = 1 r 1
- 1÷3 = 0 r 1
44₁₀ = 1122₃
B. Add two numbers
The usual rules of addition apply. A sum of two digits that equals or exceeds the value of the base causes a carry to the next digit to the left. (It can be convenient to write the carries on a separate line.)
In the first attachment, the sum of the two numbers above is developed in base 3. The relevant addition facts for base 3 are ...
Of course, zero (0) is still the additive identity element.
C. Subtract two numbers
The usual rules of subtraction apply. When a larger digit is subtracted from a smaller one, a "borrow" is indicated from the digit immediately to the left. The relevant subtraction facts for base 3 are ...
- 0 - 2 = 1 (borrow 1)
- 0 - 1 = 2 (borrow 1)
- 1 - 2 = 2 (borrow 1)
- 1 - 1 = 0
- 2 - 2 = 0
- 2 - 1 = 1
Example: 67 -44 = 23, done in base 3
For subtraction of a smaller number from a larger one, it can work well to do the computation left to right:
2111 -1122 = ...
4th digit: 2 - 1 = 1
3rd digit: 1 - 1 = 0
2nd digit: 1 - 2 = 2 (borrow 1 from 3rd digit: 0 - 1 = 2, borrow 1 from 4th digit: 1 - 1 = 0). The difference is now 0 2 2 x
1st digit: 1 - 2 = 2 (borrow 1 from 2nd digit: 2 - 1 = 1). The difference is now ...
2111 -1122 = 0212, done in base 3
D. Rationale
Consider the expanded forms of the sum and difference numbers in base 3:
11010₃ = 1×3⁴ +1×3³ +0×3² +1×3¹ +0×3⁰ = 81 +27 +3 = 111₁₀ (67+44=111)
212₃ = 2×3² +1×3¹ +2×3⁰ = 18 +3 +2 = 23₁₀ (67-44=23)
These forms clearly show the place value of a digit position is given by the corresponding power of the number base. You see this when you write base-10 numbers in expanded form. The point is driven home by working with numbers in other bases.
In short, working with numbers in other bases gives you a better understanding of place value.
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Additional comment
Fractions can be converted to a different base by multiplying the fraction by the base, then using the integer digit as the next digit to the right.
Consider 5/9 in base 3:
- 5/9 × 3 = 1 2/3
- 2/3 × 3 = 2
Then 5/9 in base 3 is 0.12
As with base 10, any fraction that cannot be written using a power of the base as its denominator will have a repeating fractional part. For example, ...
1/10 = 0.00220022...(4-digit repeat) in base 3