GRE Math from Scratch

Integers

Complete positive and negative numbers and 0. They do not include fractions or decimals. {…, -3, -2, -1, 0, 1, 2, 3, …}

When integers are added, subtracted, or multiplied, the result is always a(n)…

integer.

The product of two positive integers is a

positive integer.

The product of two negative integers is a

positive integer.

The product of a positive integer and a negative integer is a

negative integer.

Factor

refers to a number that divides another number evenly in the context of multiplication.

Divisor

Any number that divides another number, potentially with a remainder.

When integers are multiplied, each of the multiplied integers is

called a factor.

Multiple

the result of multiplying factors or integers.

Divisible

when one integer can be divided by another without leaving a remainder.

Least Common Multiple

the least positive integer that is a multiple of both c and d (two nonzero integers).

The Greatest Common Divisor

The greatest positive integer that is a diviser of both c and d (two nonzero integers).

Break down both integers into their prime factorizations and multiply all the prime factors they have in common.

The Greatest Common Divisor is also called…

The greatest common factor

Quotient

the result of dividing one number by another, or the number of times a divisor can be evenly subtracted from a dividend.

Remainder

The amount left over after division when one integer is divided by another.

Even Integer

An integer that is divisible by 2 without any remainder.

Odd Integer

An integer that is not divisible by 2, resulting in a remainder of 1 when divided by 2.

The sum of two even integers =

An even integer.

The sum of two odd integers =

An even integer.

The sum of an even integer and an odd integer =

an odd integer.

The product of two even integers =

an even integer.

The product of two odd integers =

an odd integer.

The product of an even integer and an odd integer =

an even integer.

Prime Number

An integer greater than 1 that has only two positive divisors: 1 and itself.

2

The only even prime number.

3

The smallest odd prime number.

5

The first prime number after 3, it is also the only prime that ends in 5.

7

The fourth prime number, it is also the sum of the first three prime numbers.

11

The fifth prime number, it is the first prime that is greater than 7.

13

The sixth prime number, it is also the sum of the first four prime numbers.

17

The sixth prime number and the first prime number greater than 13.

19

The eighth prime number, it follows 17 and precedes 23.

23

The ninth prime number, it follows 19 and precedes 29.

29

The tenth prime number, it follows 23 and precedes 31.

First 10 Prime Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, and 29

Prime Factorization

The process of expressing a number as the product of its prime factors.

Composite Number

An integer greater than 1 that is not a prime number.

4

is the smallest composite number.

6

The smallest composite number, which can be expressed as 2 x 3.

8

is the next composite number after 6.

9

is the smallest odd composite number.

10

The smallest two-digit composite number, which can be expressed as 2 x 5.

12

is the smallest even composite number greater than 10, which can be expressed as 3 x 4.

14

is an even composite number and can be expressed as 2 x 7.

15

The smallest odd composite number greater than 10, which can be expressed as 3 x 5.

16

is the smallest even composite number greater than 14, which can be expressed as 4 x 4.

18

is an even composite number that can be expressed as 2 x 9 or 3 x 6.

First 10 Composite Numbers

4, 6, 8, 9, 10, 12, 14, 15, 16, 18

Fraction

A number of the form ^c/_d , where c and d are integers and d ≠ 0

Numerator

The integer c on the top of the fraction.

Denominator

The integer d on the bottom of the fraction.

Rational Numbers

numbers that can be expressed as a fraction ^c/_d, where c and d are integers, and d is not zero. In simpler terms, they are numbers that can be written as a ratio of two whole numbers.

Every integer n is a…

rational number (because n = ⁿ/₁)

^-c/_d =

^c/_-d

^c/_-d =

-^c/_d

Common Denominator

A common multiple of the two denominator used to add or subtract two fractions with different denominators.

Mixed Number

An expression that consists of an integer part and a fraction part, where the fraction part has a value between 0 and 1.

Meaning of Mixed Number 4³/₈

4 + ³/₈

To convert a mixed number to a fraction

convert the integer part to an equivalent fraction with the same denominator as the fraction, and then add it to the fraction part.

Fractional Expressions

Numbers of the form ^c/_d , where either c or d is not an integer and d ≠ 0. Can be manipulated just like fractions.

Exponents

Used to denote the repeated multiplication of a number by itself. The base is the number being repeated and the exponent is the number of times it is being multiplied by itself.

When negative numbers are raised to powers…

The result may be positive or negative.A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative.

A negative number raised to an even power is

always positive.

A negative number raised to an odd power is

always negative.

Negative number outside of parentheses raised to a positive or a negative power is

always negative.

-3² = - ((3)(3)) = -9

-3³ = - ((3)(3)(3)) = -27

For all nonzero numbers a, a⁰ =

1

For all nonzero numbers a, a^-1 =

¹/_a

For all nonzero numbers a, a^-2 =

¹/_a²

(a) (a^-1) =

(a) (¹/_a) = 1

Square Root of a nonnegative number n is

a number r such that r² = n

All positive numbers have…

Two square roots, one positive and one negative.

Square root of 0 =

0

The Square Root symbol placed over a nonnegative number denotes

The nonnegative/positive square root.

Square roots of negative numbers are

not defined in the real number system.

(√a)² =

a

√a² =

a

√a √b =

√ab

^√a/_√b

√^a/_b

Odd Order Roots

Roots taken of odd numbers (e.g., 3/cube root or 5). The sign of the number n is preserved.

Even Order Roots

Roots taken of even numbers (e.g., squareroot, fourth root).

For odd order roots, there is exactly x root(s) for every number n (³√n), even when n is negative.

one

For even order roots, there are exactly x root(s) for every positive number n (√n) and y root(s) for any negative number n.

two; zero

Decimals

A number system is based on representing numbers using powers of 10. The place value of each digit corresponds to a power of 10.

The number 7,532.418 can be written as

7(1,000) + 5(100) + 3(10) + 2(1) + 4(¹/₁₀) + 1(¹/₁₀₀) + 8(¹/_1,000)

OR

7(10³) + 5(10²) + 3(10¹) + 2(10⁰) + 4(10^-1) + 1(10^-2) + 8(10^-3)

If there are a finite number of digits to the right of the decimal point, every decimal can be…

converted to an integer divided by a power of 10 (since each place value is a power of 10).

2.3 =

2 + ³/₁₀

= ²⁰/₁₀ + ³/₁₀

= ²³/₁₀

Every fraction with integers in the numerator and denominator can be converted to an equivalent decimal by

dividing the numerator by the denominator using long division.

The decimal that results from the long division will either terminate or repeat without end.

Every rational number can be expressed as…

a terminating or repeating decimal.

Irrational Numbers

Real numbers that cannot be expressed as a fraction ^c/_d, where c and d are integers and d is not zero. They are non-terminating and non-repeating decimals.

Real Numbers

Consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line.

Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive.

Interval

2 < x < 3

A real number x is between 2 and 3 on the number line.

The set of all real numbers that are between 2 and 3.

The entire real number line is also considered to be an interval.

Absolute Value |x|

The distance between a a number x and 0 on the number line.

Property of Real Numbers:

r + s =

s + r

Property of Real Numbers:

rs =

sr

Property of Real Numbers:

(r + s) + t =

r + (s + t)

Property of Real Numbers:

(rs)t =

r(st)

Property of Real Numbers:

r(s + t) =

rs + rt

Property of Real Numbers:

r + 0 =

r

Property of Real Numbers:

(r) (0) =

0

Property of Real Numbers:

(r) (1) =

r