GMAT Exam: A Simple Guide To Arithmetic - Numbers and Operations

Counting Numbers, Whole Numbers, and Integers
The counting numbers (or natural numbers) are the numbers 1, 2, 3, and so on. The whole numbers are the numbers 0, 1, 2, 3, and so on. Whole numbers that are greater than 1 are either prime or composite. A prime number is a whole number greater than 1 that has exactly two distinct whole number factors: itself and 1. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The whole numbers greater than 1 that are not prime are the composite numbers. They are the numbers that have more than two distinct whole number factors. The number 1 is neither prime nor composite.

The integers are either positive (1, 2, 3, …) or negative (… –3, –2, –1) or zero.

Note: The three dots indicate that the pattern continues without end. On the number line, positive numbers are located to the right of zero and negative numbers are to the left of zero. Zero is neither positive nor negative:



Integers that divide evenly by 2 are even. The even integers are …–6, –4, –2, 0, 2, 4, 6,.… Integers that do not divide evenly by 2 are odd. The odd integers are … –5, –3, –1, 1, 3, 5,.…

Tip: Notice that 0 is an even integer.

Decimals
Decimals are written using a base-10 place-value system. The value of a decimal number is determined by the placement of the decimal point in the number. A place-value diagram for some of the positional values of the decimal system is shown here.



In a whole number, the decimal point is understood to be to the immediate right of the rightmost digit.

Fractions
The fraction is composed of three parts. The line between n and d is the fraction bar. The number above the fraction bar is the numerator, and the number below the fraction bar is the denominator. The fraction means n ÷ d. Tip: The denominator of a fraction can never be zero, because division by zero is undefined.

The reciprocal of the fraction is the fraction , provided a ≠ 0 and b ≠ 0.

In proper fractions (such as ), the numerator is less than the denominator.

In improper fractions (such as and ), the numerator is greater than or equal to the denominator.

The value of an improper fraction is greater than or equal to one.

A mixed number (such as ) is the sum of a whole number and a fraction.

Change an improper fraction to a mixed number or a whole number by performing the indicated division.

If there is a remainder, write it as the numerator of a fraction that has the divisor as the denominator, for example:


Change a mixed number to an improper fraction by multiplying the whole number part by the denominator of the fractional part.

Then add the numerator of the fractional part to the resulting product.

Place the resulting sum over the denominator of the fractional part, for example:


If both the numerator and denominator of a fraction are multiplied or divided by the same nonzero number, the value of the fraction is unchanged.

The resulting fraction is equivalent to the original fraction.

For example, and are equivalent fractions.

Similarly, and are equivalent fractions.

To write two fractions as equivalent fractions with the same denominator, use the least common multiple as the common denominator.

A multiple of a number is the product of the number and any counting number. The least common multiple (or LCM) of two numbers is the least counting number that is a multiple of both numbers. List multiples of the greater number, in order.

Stop when you first list a multiple that is also a multiple of the other number.

This multiple will be the LCM of the two numbers.

For instance, to write and as fractions with the same denominator, find the LCM (20, 15).

List the multiples of 20, in order: 20, 40, 60. Stop.

The number 60 is the first multiple of 20 that is also a multiple of 15, so 60 is the LCM (20, 15). Thus,


When the only common factor between the numerator and denominator of a fraction is one, the fraction is in lowest terms.

Reduce a fraction to lowest terms by dividing the numerator and denominator by their greatest common factor.

The greatest common factor (or GCF) of two numbers is the largest factor common to the two numbers.

Tip: The GCF is always positive.

For example, reduce by dividing its numerator and denominator by the GCF of 60 and 84.

List the positive factors of 60 in pairs:


List the positive factors of 84 in pairs:


To read the factors of a number from least to greatest: Start at the top left of the factor table, move left to right across the top row, and then right to left across the bottom row.

Examination of the two lists shows GCF (60, 84) is 12.

Thus,


To obtain the equivalent decimal representation of a fraction, such as , perform the indicated division.

Divide the numerator by the denominator. Insert a decimal point in the numerator and one or more zeros (as needed) to the right of the decimal point to complete the division:


Percents
Percent means “per hundred.”

The % sign stands for or 0.01.

To change a percent to an equivalent fraction, substitute multiplying by for the percent sign.

For example, .

To change a percent to an equivalent decimal, substitute multiplying by 0.01 for the percent sign.

For example, 25% = 25(0.01) = 0.25.

Rational and Irrational Numbers
All of the counting numbers, whole numbers, integers, fractions, decimals, and percents are rational numbers.

The rational numbers are numbers that can be expressed as , where p and q are integers (q ≠ 0).

In other words, a rational number is a number that can be expressed as a quotient of an integer divided by an integer other than zero.

Tip: Zero is excluded as a denominator for because division by zero is undefined, so has no meaning no matter what number you put in the place of p.

The decimal representations of rational numbers terminate or repeat a block of digits.

For instance, is a rational number whose decimal representation terminates, and … is a rational number whose decimal representation repeats the block of digits “18.”

In most problems, you can round repeating decimals to a certain number of decimal places.

For instance, rounded to two decimal places, is approximately 0.18.

The irrational numbers are numbers whose decimal representations neither terminate nor repeat.

These numbers cannot be expressed as the quotient of two integers.

For instance, the positive number that multiplies by itself to give 2 is an irrational number called “the positive square root of 2.”

You use the square root symbol to show the positive square root of 2 like this: .

You cannot express as the quotient of two integers, nor can you express it precisely in decimal form.

Its decimal equivalent continues on and on without a pattern of any kind.

No matter how far you go with decimal places, you can only approximate .

For instance, rounded to three decimal places, is approximately 1.414.

There are infinitely many square roots and other roots as well that are irrational.

Real Numbers
The real numbers are all the rational and irrational numbers put together. All numbers used on the GMAT are real numbers. The relationship of the various sets of numbers included in the real numbers is shown here.



Real numbers are sometimes called signed numbers because they are positive, negative, or zero.

Positive numbers lie to the right of zero on the number line and negative numbers lie to the left of zero.

Zero is neither positive nor negative. It has no sign.

The absolute value of a real number is its distance from zero on the number line. The absolute value of a nonzero real number is positive. The absolute value of zero is zero. For example, the absolute value of –3.4 is 3.4, written as |–3.4| = 3.4.

Every real number has an opposite. If a real number is positive, its opposite is negative. If a real number is negative, its opposite is positive.

For example, the numbers and are opposites. Zero is its own opposite. A number and its opposite have the same absolute value.

For instance,


Operations with Real Numbers
Addition, subtraction, multiplication, and division are the four basic arithmetic operations. Following are rules that you will find useful for completing basic operations:

Adding Fractions


Subtracting Fractions


Multiplying Fractions

The process of multiplying fractions can be simplified by dividing out common factors, if any, before doing any multiplication.

For example, .

Also, remember you do not have to find a common denominator when multiplying fractions.

Dividing Fractions


Adding Decimals


Subtracting Decimals


Multiplying Decimals


Dividing Decimals


Adding and Subtracting Real Numbers


Multiplying and Dividing Real Numbers


Properties of Operations with Real Numbers
The following 11 properties hold for the operations of addition and multiplication for all real numbers a, b, and c:

- Closure Property of Addition: (a + b) is a real number. This property guarantees that the sum of any two real numbers is always a real number.

- Closure Property of Multiplication: (a·b) is a real number. This property guarantees that the product of any two real numbers is always a real number.

- Commutative Property of Addition: a + b = b + a. This property allows you to reverse the order of the numbers when you add, without changing the sum.

- Commutative Property of Multiplication: a·b = b·a. This property allows you to reverse the order of the numbers when you multiply, without changing the product.

- Associative Property of Addition: (a + b) + c = a + (b + c). This property says that when you have three numbers to add together, the final sum will be the same regardless of the way you group the numbers (two at a time) to perform the addition.

- Associative Property of Multiplication: (ab)c = a(bc). This property says that when you have three numbers to multiply together, the final product will be the same regardless of the way you group the numbers (two at a time) to perform the multiplication.

The associative property is needed when you have to add or multiply more than two numbers because you can do addition or multiplication on only two numbers at a time. Thus, when you have three numbers, you must decide which two numbers you want to start with—the first two or the last two (assuming you keep the same order). Either way, your final answer is the same.

- Additive Identity Property: There exists a real number 0, called the additive identity, such that a + 0 = a and 0 + a = a. This property guarantees that you have a real number, namely 0, for which its sum with any real number is the number itself.

- Multiplicative Identity Property: There exists a real number 1, called the multiplicative identity, such that a·1 = a and 1·a = a. This property guarantees that you have a real number, namely 1, for which its product with any real number is the number itself.

- Additive Inverse Property: For every real number a, there is a real number called its additive inverse, denoted –a, such that a + –a = 0 and –a + a = 0. This property guarantees that every real number has an additive inverse (its opposite) that is a real number whose sum with the number is 0.

- Multiplicative Inverse Property: For every nonzero real number a, there is a real number called its multiplicative inverse, denoted a–1 or , such that and . This property guarantees that every real number, except zero, has a multiplicative inverse (its reciprocal) whose product with the number is 1.

Notice that when you add the additive inverse to a number, you get the additive identity (0) as an answer, and when you multiply a number by its multiplicative inverse, you get the multiplicative identity (1) as an answer.

- Distributive Property: a(b + c) = a·b + a·c and (b + c)a = b·a + c·a. This property says that when you have a number times a sum (or a sum times a number), you can either add first and then multiply, or multiply first and then add. Either way, the final answer is the same.
Notice that the distributive property involves both addition and multiplication at the same time. Another way to express the distributive property is to say that multiplication distributes over addition.