Comparision in General
On a number line, a number or point may be to the left or right of
another. A a first number a is to the right of a second number b when
the first number = the second number + a positive number
For example,
+5 is 3 to the right of +2. Here it +5 is 3 more than +2 +5 is 5 to the
right of 0. Here it is +5 is 5 more than 0.
+5 is 7 to the right of -2. Here it is +5 is 7 more than 0.
-3 is 1 to the right of -4. Here -3 is 1 more than -4.
-3 is 7 to the right of -10. Here -3 is 7 more than -10.
Method
To compare two numbers (or two expressions) a first and second,
calculate the
difference = first number - second number
The difference will be positive or negative or zero.
If the difference first number - second number is positive,
then the first number is to the right of the second. The first number is
more than the second by the difference. Using the more than symbol
>, we may write
first number > second number
Or, we may say and write the second number is less than the first by
the difference. Using the less than symbol <, we may write
second number < first number
Note: The number of units between the two numbers is given by the
difference, its unsigned part.
If the other difference second number - first number is positive,
then
The second number is more than the first by the difference. Using the
more than symbol >, we may write
second number > first number
Or, we may say and write the first number is less than the second by
the difference. Using the less than symbol <, we may write
first number < second number
Note: The number of units between the two numbers is given by the
difference, its unsigned part.
In practice, the two differences
- first number - second number
- second number - first number
have opposite signs and the same unsign part or length - or magnitude
or absolute value. Why is an exercise in algebraic reasoning. The common
length gives the number of units or distance between the two numbers
in the number line - a 1D coordinate axis.
Changing the Language
Avoid the name greater than for >
Use
the name more than instead
In the previous lessons, we have seen how the more than phrase
can be used in all circumstance where a first number is more than another
by a positive amount. Now the more than sign > is also
called the greater than sign. In the comparision of
counts and measures, and positive
numbers, a first number is more than a second when the length
of the first is longer, bigger and greater than the length of the second.
Before people learn about negative numbers, the greater than concepts represents
a comparision of size or magnitude or length. We may read - 3 > -5 by 2 as
-3 more than -5 by 2 units, current or previous mathematics language
read - 3 > -5 as -3 greater than -5 with the technical understanding that
the greater than phrase here means more than and does not
include or mean a comparision of length magnitudes. Clearly, -3 with
unsigned part 3 units which is smaller than 5, the unsigned part of -5. Professor
Whyslopes, this site author, would like to avoid the confusion here between
the technical use of the name greater than for the the sign >, and
and common meaning of greater name as a comparison of size or magnitude by
changing the language, so that the sign >s is called the more than sign.
Fortunately, a similar conflict for the less than sign < would have appeared
if the latter had been called the smaller than sign.
Why Say "More Than" Instead of "Greater Than"
The concept of greater than or more than is understood by
students when dealing with counts or unsigned whole numbers. Before the
introduction of signs, that is negative and positive numbers, finite
decimal expansions extend this idea of greater than or more
than. A finite decimal expansion in particular counts the number of
units, tenths, thousandths and so on that the number it represents can be
divided into.
With the introduction of positive and negative numbers and zero on say
the real number line, the technical ideas of greater than differs
from the common usage, or the introductory idea of comparison of by size
or magnitude (apart from any signs that may be present). Because of this
students are tempted to say that a real number a is greater than
another real number b if the magnitude of a is greater than
the magnitude of b. The latter means real number a is
greater in magnitude than another real number b.
The task is to remove the temptation or conflict. The symbol >
traditional has been called the greater than sign. Technically, given two
real numbers a and b we write a>b if and only if
there is positive number c such that a = b + c. So a
is c units more than b.
To avoid confusion, and to align mathematical terminology with the common
usage, the symbol > should be named or renamed the
more positive than symbol on first usage in mathematics. This new
name corresponds precisely to the technical meaning. With this new
convention, the phrase a greater than b can revert to the
common usage and mean |a| > |b|, a comparison of magnitude.
Similarly, a < b can be read not as a is less than
b but as a is more negative than b.
This new terminology means there is a positive number c such
that a = b - c or equivalently such that a + c = b. The
signs <= and >= now may be read as more negative
or equal to and more positive or equal to.
Linear and Nonlinear Orderings (optional)
A number b is said to between two other numbers a and
c if and only if there is a positive number q < 1 such
that b=qa + (1-q)c.
Ordering of the real line by the relationship more positive than
provides a linear ordering of the real line: for any three points
a, b and c on the real line the relationships a
< b <c imply that b is between a and c.
Ordering by magnitude provides a linear ordering of the positive numbers.
For any three unsigned (positive) numbers a, b and
c, the relationship a < b < c implies that b
is between a and c. But for any three points a,
b and c on the real line the relationship |a| < |b|
< |c| does not imply that b is between a and
c. So ordering by magnitude (or absolute value) of points on the
whole real line is nonlinear.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
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Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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