Division is the inverse (or the opposite) of multiplication. Dividing the product by any of its factor will yield its complement (the other factors).
Terms of division can also be arranged into a fraction, with the dividend becoming the numerator and the divisor the denominator. These terms may still be divided to yield a quotient.
Any value that can be simplified to a fraction of an integer over one, is itself an integer. Likewise, any integer can be represented as a fraction with a denominator of 1.
Zero divided by any value (that is not zero) is equal to zero.
Any value divided by zero is not a real number.
The reciprocal of a value is itself divided by one. Any value multiplied by its reciprocal equals the multiplicative identity (1).
If both the numerator and denominator have the same sign (positive or negative), then the quotient is positive.
If the numerator and denominator have different signs, then the quotient is negative.
To sum 2 fractions: first, get both fractions over the same denominator; then, add the numerators over the common denominator.
To difference 2 fractions: first, get both fractions over the same denominator; then, subtract the numerators over the common denominator.
To multiply 2 fractions, simply multiply the numerators and denominators together respectively.
To divide 2 fractions, multiply the dividend with the reciprocal of the divisor.
The quotient between a integer and the following values is also an integer if the dividend has the respective characteristics.
1: all real numbers
2: ends with an even digit
3: sum of all digits divided by 3 is an integer
4: the last two (right-most) digits divided by 4 is an integer
5: last digit is a 0 or 5
6: the value is an integer when divided by 2 and 3 (shares both characteristics)
7: no simple rules exist
8: the last three digits divided by 8 is whole
9: sum of digits divided by nine is whole
10: ends with a zero
Long division is used to divide values that are too large to do mentally.
Any time the numerator, or dividend is zero; then, the quotient is zero.
Any time the denominator, or divisor is zero; then the quotient is not a member of the real number set (not real).
Any time that the numerator and denominator are the same value, the quotient will equal 1.
Any time the numerator and denominator have different signs (one is positive while the other is negative), the quotient will be negative.
We can reduce a fraction by removing similar quantities from the top and the bottom iff (if and only if):
If all 3 conditions are true, we may remove the two numbers from the fraction.
By using the other properties of arithmetic, we may rearrange the fraction to get the similar values on the top and bottom.
A fraction is considered reduced when no common factors exist in the numerator and denominator.
When multiplying two fractions together, multiply straight across (numerator with numerator, denominator with denominator). When dividing fractions, flip the second then multiply.
Remember that any integer can be represented as a fraction of itself over one.
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