The Meaning of a|b

Don’t rush into the conclusion that [latex]\large{a|b}[/latex] is equivalent to the rational number [latex]\LARGE{a \over b}[/latex].

The truth is, the mathematical notation [latex]\large{a|b}[/latex] is NOT the same as the fraction [latex]\LARGE{a \over b}[/latex].

The vertical line or bar , |, between [latex]{a}[/latex] and [latex]{b}[/latex] is called the pipe.

The notation [latex]\color{red}{a|b}[/latex] is read as “[latex]a[/latex] divides [latex]b[/latex]”.

pipe is the name of the vertical bar in a|b; and a|b is also a shorthand for a divides b

NOTE: The assumption here is that [latex]{a}[/latex] and [latex]{b}[/latex] are integers but [latex]{a}[/latex] is not equal to zero, [latex]a \ne 0[/latex]. Notice, if we allow or assign the variable [latex]a[/latex] to take the value of zero, we will have a situation where the integer [latex]b[/latex] is being divided by [latex]0[/latex] which is undefined.

Expressing a|b in Equation Form

Interestingly, the math notation [latex]\color{red}{a|b}[/latex] can be expressed as an equation, which will help us make more sense of it.

Let’s talk about it for a moment. If [latex]a[/latex] divides [latex]b[/latex], it implies that [latex]a[/latex] can evenly divide [latex]b[/latex]. Therefore, when integer [latex]b[/latex] is divided by integer [latex]a[/latex], it doesn’t leave a remainder which suggests that the remainder is zero, [latex]0[/latex].

For instance, we know that 2 evenly divides 10, wherein the quotient is 5 with a remainder of zero.

Notice that we can rewrite the equation on the left as [latex]10 = 2 (5)[/latex] which precisely implies that the dividend is equal to the product of the divisor and the quotient.

Study carefully the two diagrams below. Pay attention especially to the placements of the dividend, divisor, and quotient of the equation on the left.

in 10/2 = 5, 10 is the dividend, 2 is the divisor while 5 is the quotient

Then compare them to the new placements or locations of the dividend, divisor, and quotient of the equation on the right.

in 10=2(5), 10 is the dividend, 2 is the divisor while 5 is the integer quotient

Observation: The dividend stays on the left side of the equation while the divisor and quotient become the first factor and the second factor respectively on the right side of the equation. This observation is going to be very useful when talking about how to convert the notation [latex]\color{red}{a|b}[/latex] into the equation form.

From Notation to Equation

Below is a useful diagram that shows how to convert the notation [latex]{a|b}[/latex] into an equation.

a divides b implies b=a(c) where a is not equal to zero and c is just some integer

DEFINITION: Suppose [latex]a[/latex] and [latex]b[/latex] are integers but [latex]a \ne 0[/latex]. We say that [latex]a[/latex] divides [latex]b[/latex] in the notation [latex]{a|b}[/latex] if there exists an integer [latex]c[/latex] such that [latex]b = a\,c[/latex] .

The following are some examples that showcase how the notation a|b can easily be written in the form of b=a(c) where c is just another integer.

[latex]\color{blue}1|2 \to 2 = 1\left( 2 \right)[/latex]

[latex]\color{blue}7|0 \to 0 = 7\left( 0 \right)[/latex]

[latex]\color{blue}5|30 \to 30 = 5\left( 6 \right)[/latex]

[latex]\color{blue}8|104 \to 104 = 8\left( {13} \right)[/latex]

[latex]\color{blue}4|32 \to 32 = 4\left( 8 \right)[/latex]

[latex]\color{blue}9|135 \to 135 = 9\left( {15} \right)[/latex]

[latex]\color{blue}3|{}^ – 15 \to {}^ – 15 = 3\left( {{}^ – 5} \right)[/latex]

[latex]\color{blue}2|{}^ – 14 \to {}^ – 14 = 2\left( {{}^ – 7} \right)[/latex]

[latex]\color{blue}6|{}^ – 102 \to {}^ – 102 = 6\left( {{}^ – 17} \right)[/latex]