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BASIC PROPERTIES OF ZERO AND ONE
Jump right to the exercises!
Zero ( 0 ) and one ( 1 ) are very special numbers.
This exercise summarizes their important properties.
Addition Property of Zero:
Adding zero to a number does not change it:
For all numbers x , x + 0 = 0 + x = x .
Multiplication Property of Zero:
Multiplying a number by zero always gives zero:
For all numbers x , x · 0 = 0 · x = 0 .
Multiplication Property of One:
Multiplying a number by one does not change it:
For all numbers x , x · 1 = 1 · x = x .
Recall that 25 (read as "2 to the fifth power") is a shorthand for 2 · 2 · 2 · 2 · 2 .
Powers of One:
The number one, raised to any power, equals one:
For all numbers n , 1n = 1 .
(Even though you may not know about negative and fractional powers yet, don't worry!
Just start getting used to the fact that one, raised to any power, is always one!)
Powers of Zero:
The number zero, raised to any allowable power, equals zero:
For n = 1, 2, 3, … , 0n = 0 .
Zero to the zero power ( 00 ) is not defined.
Ways to denote division:
A number x divided by a number y can be denoted in several ways:
x ÷ y or x/y or
xy .
The last way (the horizontal fraction) is preferred.
In all these forms, x is called the numerator and
y is called the denominator.
Ways to denote multiplication:
A number x multiplied by a number y can be denoted in several ways:
x · y (using a vertically centered dot), or
xy (using juxtaposition), or
(x)(y) (using parentheses).
In algebra and beyond, do NOT use the symbol × to denote multiplication, since
it can be confused with the variable x .
Zero as a numerator:
Zero, divided by any nonzero number, is zero:
For all x&neq;0 ,
0x=0 .
Division by zero is not allowed:
Any division problem with zero as the denominator is not defined:
For example,
00 ,
20 , and
5.70 are not defined.
Names for the number one:
Any nonzero number divided by itself equals one:
For all x&neq;0 ,
xx=1 .
Names for the number zero:
The numbers 3 and -3 are opposites;
they are the same distance from zero, but on opposite sides of zero.
Any number added to its opposite is zero:
For all numbers x ,
x+(-x)=
(-x)+x=0 .
Note: the "opposite of x" is also called the "additive inverse of x";
it is the number which, when added to x, gives zero.
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.