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Note \(\sqrt{negative numbers}\) or \(\frac{number}{0-zero}\) are
NOT real !!
On the GRE by default, any number mentioned in the quant section is a
real number so must NOT assume that it is an integer unless stated clearly.
1) The Number LineA real number line, or simply number line, allows us to visually display real numbers by associating them with unique points on a line. The real number associated with a point is called a coordinate12. A point on the real number line that is associated with a coordinate is called its graph. To construct a number line, draw a horizontal line with arrows on both ends to indicate that it continues without bound. Next, choose any point to represent the number zero; this point is called the origin.
The scale need not always be one unit. Each tick mark might represent a unit fraction.
The opposite of any real number a is \(−a\). Opposite real numbers are the same distance from the origin on a number line, but their graphs lie on opposite sides of the origin and the numbers have opposite signs.
The opposite of a negative number is positive: \(−(−a) = a\).
2) Absolute ValueThe absolute value of a real number a, denoted |a|, is defined as the distance between zero (the origin) and the graph of that real number on the number line. Since it is a distance, it is always positive.
\(|-4|=4\) and \(|4|=4\)
Also, it is worth noting that \(|0|=0\)
Think of a real number whose distance to the origin is 5 units. There are two solutions: the distance to the right of the origin and the distance to the left of the origin, namely, {±5}. The symbol (±) is read “plus or minus” and indicates that there are two answers, one positive and one negative.
\(|-5|=5\) and |\(5|=5\)
See more here:
Absolute NumbersTip
If there is an even number of consecutive negative signs, then the result is positive. If there is an odd number of consecutive negative signs, then the result is negative.
KEY TAKEAWAYS
• Any real number can be associated with a point on a line.
• Create a number line by first identifying the origin and marking off a scale appropriate for the given problem.
• Negative numbers lie to the left of the origin and positive numbers lie to the right.
• Smaller numbers always lie to the left of larger numbers on the number line.
• The opposite of a positive number is negative and the opposite of a negative number is positive.
• The absolute value of any real number is always positive because it is defined to be the distance from zero (the origin) on a number line.
• The absolute value of zero is zero.
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