Zero and One: The Fundamental Pillars of Mathematics

Step-by-step Guide to Understand Properties of Zero and One

Properties of Zero

1. Multiplication with Zero: For any real number $a$, the product of $a$ and zero is always zero ($a×0=0$). This property is crucial because it underlines that multiplying any number by zero results in zero.
2. Addition with Zero: For any real number $a$, adding zero to $a$ does not change $a$ ($a+0=a$). Similarly, $a−0=a$. This is known as the identity property of addition, where zero is the additive identity.
3. Subtraction with Zero: Subtracting zero from any real number $a$ leaves $a$ unchanged ($a−0=a$). However, when zero is subtracted from $a$ ($0−a$), the result is $−a$, the additive inverse of $a$.
4. Division by a Nonzero Number: For any nonzero real number $a$, dividing zero by $a$ results in zero ($0÷a=0$). This is because zero divided by any number is always zero.
5. Zero Product Property: If the product of two real numbers $a$ and $b$ is zero ($a×b=0$), then at least one of the numbers must be zero. This property is fundamental in solving quadratic equations.

Properties of One and Minus One

1. Multiplication with One: The product of one and itself is one ($1×1=1$). This demonstrates that one has a multiplicative identity.
2. Multiplication with Minus One: Multiplying minus one with itself gives one ($(−1)×(−1)=1$), while multiplying minus one with one gives minus one ($(−1)×1=−1$).
3. Identity Property of Multiplication: For any real number $a$, multiplying $a$ by one leaves it unchanged ($a×1=a$).
4. Multiplying with Minus One: Multiplying any real number $a$ by minus one gives the additive inverse of $a$ ($a×(−1)=−a$).
5. Distributive Property of Minus One: For all real numbers $a$ and $b$, multiplying the product of $a$ and $b$ by minus one gives the same result as multiplying $a$ by minus one and then by $b$, or $a$ by $b$ and then by minus one ($−1(a×b)=(−a)×b=a×(−b)$).
6. Negation of Negation: For all real numbers $a$, multiplying minus one by the additive inverse of $a$ returns $a$ ($−1×(−a)=a$).
7. Division by One: For any real number $a$, dividing $a$ by one results in $a$ itself ($a÷1=a$), emphasizing that one is the multiplicative identity.
8. Division by Minus One: Dividing any real number $a$ by minus one gives the additive inverse of $a$ ($a÷(−1)=−a$).
9. Successive Integers: For any integer $n$, adding one to $n$ yields the next larger integer, and adding minus one to $n$ gives the next smaller integer.
10. Counting and Inverse Addition: The smallest counting number is $1$. Adding one and its additive inverse (minus one) results in zero ($1+(−1)=0$).
11. Zero Sum Property: For any real number $a$, the sum of $a$ and its additive inverse is zero ($(1×a)+(−1×a)=a+(−a)=0$).

These properties are vital in algebra and arithmetic, forming the basis for various mathematical operations and problem-solving techniques. Understanding and applying these properties can greatly simplify calculations and conceptual understanding in mathematics.

Examples:

Example 1:

If $x$ and $y$ are two numbers such that $x×y=0$, and $x=5$, what is $y$?

Solution:
According to the zero product property, if the product of two numbers is zero, then at least one of them must be zero. In this case, since $x×y=0$ and $x=5$, it means that $y$ must be $0$. Therefore, $y=0$.

Example 2:

If $a$ is any real number, and you have the equation $a×1+(−1)×a=b$, what is the value of $b$?

Solution:

First, apply the identity property of multiplication, which states that any number multiplied by one is itself. So, $a×1=a$.

Next, multiplying a number by $−1$ gives its additive inverse. So, $(−1)×a=−a$.

Therefore, the equation becomes $a+(−a)=b$.

Since $a+(−a)$ equals zero (as adding a number and its additive inverse results in zero), we get $b=0$.