Math is filled with theorems, axioms, and properties. Though the following properties may seem simplistic, the ISEE will couple these properties with difficult concepts, such as algebraic functions. It is important to understand the following properties thoroughly.
This is useful because even when you do not know the individual values (such as in an algebraic equation), you know a value is equal to itself.
If one value equals a second value and the second value equals a third value, then the third value equals the first value.
For example, if you know:
19 + ▢ = 5 + 14 and 5 + 14 = 19
Then you also know that: 19 + ▢ = 19
Thus, ▢ must be 0.
When you are only adding or multiplying values, you may change the order without changing the result.
4 + 7 + 9 + ▢ = 6 + 7 + 4 + 9
We can rearrange the right side of the equation:
4 + 7 + 9 + ▢ = 4 + 7 + 9 + 6
Now, it is easy to see that ▢ = 6.
This example would be exactly the same if we changed all the plus signs to multiplication signs.
When you are only adding or multiplying values, you may change the groupings of values without changing the result. The groups are separated by parentheses. For example:
You could add different parentheses on each side of the equation, but this will not change the equivalency. In the following case, the parentheses merely changes the order of addition:
The same is true for multiplication:
You must add the same value to both sides of the equals sign to maintain the equality of the values.
Take the following equation:
If you were to add 5 to the left side, the equation would read:
This is not a valid equation, since the two sides are not balanced. Instead, you must add 5 to both sides:
You must multiply the same value to both sides of the equals sign to maintain the equality of the values.
Same as with addition, the equation must be balanced on both sides:
If you were to multiply the left side by 3, the equation would read:
Once again, this is not a valid equation, since the two sides are not balanced. Instead, you must multiply both sides:
You may multiply a value to a sum by first multiplying the value to the sum’s addends and then adding those products together. For example
When dividing values, it is the same thing as multiplying the value in the numerator by the reciprocal of the value in the denominator. This means that multiplication and division are defined as the opposite of one another, which is shown below.
The product of any value and zero will be zero.
5 × 0 = 0
Zero divided by any value is zero. For example:
In other words, 0 divided by any number will always be 0. That is, unless you are dividing 0 by 0, which does not exist at all! This is a special case that you will learn about later on.