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MATH BASICS

CHAPTER 10

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Look at the following problem:

How many even prime numbers are there between 0 and 100.

• A. 0• B. 1• C. 2• D. 3• E. 4

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The answer is (B)

• If you know what the terms even and prime mean, then this problem is a snap. Without knowledge of the problem is impossible.

• We will begin our math review by going over all the basic terms and operations covered on the ACT.

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MATH TERMINOLOGY

• Make sure that you are familiar with math terminology. Many partial answers rely on the misinterpretation of key terms; don’t be a vicitm!

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Basic Terms: Real numbers

• Real numbers are all the number you think of when you think of numbers.

• 5, ¼, 7.9, √2, are all real numbers• They include everything except imaginary

numbers, which appear only occasionally on the ACT.

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Basic terms: rational numbers

• Any number that can be written as a whole number, a fraction, or a repeating decimal is a rational number.

• 5, 1/5, and .333 are rational numbers• Most of the numbers you’ll see on th ACT

are rational numbers

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Basic terms: irrational numbers

• An irrational number cannot be written as an integer over another integer.

• ∏ is irrational and like other irrational numbers it goes on forever.

• Other irrational numbers include any square root of a number that does not have a perfect square root.

• √3 and √2 are irrational, but √4, which simplifies to 2 is rational

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Basic terms: integers

• Integers include everything except what we normally think of as fractions or decimals.

• 2, 134, -56, 0 and 7 are all integers

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Basic terms: positive and negative

• Positive numbers are to the right of the 0 on a number line and negative numbers are to the left of the 0 on the number line.

• Zero itself is neither positive or negative• There are 3 rules for positive and negative

multiplicationpositive x positive = positive

positive x negative = negative

negative x negative = positive

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Basic terms: even and odd numbers

• Even numbers are numbers that can be divided by 2 ( with no remainder)

• Odd numbers are integers that cannot be divided evenly by 2

• NOTE that 0 is even

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Basic terms: digits

• There are ten digits: 0,1,2,3,4,5,6,7,8,9• The number 364 has three digits – 3 ,6 and

4. 4 is called the ones digit, 6 is the tens digit, and 3 is the hundreds digit.

• Other digits include tenths, digit, hundredths digit, and thousandths digit

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Basic terms: prime numbers

• A prime number can be divided evenly by two and only two distinct factors – 1 and itself. Thus 2,3,5,7,11,13 are all prime numbers

• The number 2 is the only even prime number

• Neither 0 nor 1 are prime numbers• There are no negative prime numbers

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Basic terms: Absolute value

• The absolute value of a number is the distance between that number and 0 on the number line.

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Basic terms: variables and coefficients

• In the expression 3x + 4y, the x and y are called variables because we don’t know what they are.

• 3 and 4 are called coefficients because you multiply the variables by them.

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BASIC OPERATIONS

• Knowing the rules of divisibility can be very useful on the ACT.

• The rules are as follows:• 1. A number is divisible by 2 if its ones digit

can be divided evenly by 2. In other words it is an even number.

• 2. A number is divisible by 3 if the sum of its digits can be divided evenly by 3.

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Divisibility rules continued

• 3. A number is divisible by 4 if its last two digits forms a number that is divisible by 4.

• 4. A number is divisible by 5 if its last digit is a 5 or 0.

• 5. A number is divisible by 6 if it is also divisible by 2 and 3.

• 6. A number is divisible by 8 if the number formed by its last 3 digits is divisible by 8

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Divisibility rules continues

• A number is divisible by 9 if the sum of its digits can be divided evenly by 9.

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Factors and Multiples

• A number is a factor of another number if it can be divided evenly into that number.

• A number is a multiple of that number if it can be divided evenly by that number.

• All integers have a limited number of factors and an infinite number of multiples.

• FACTORS FEW, MULTIPLES MANY

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Standard symbols

SYMBOLS MEANING

= Is equal to

≠ Is not equal to

< Less than

> Greater than

≤ Less than or equal to

≥ Greater than or equal to

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Exponents

• An exponent is a short hand way of writing the value of a number multiplied several times by itself.

• The larger number is called the base and the upper number is called the exponent.

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Multiplying numbers with the same base

• When you multiply numbers that have the same base, you simply add the exponents

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Dividing numbers with the same base

• When dividing exponents with the same base, you simple subtract the bottom exponent from the top exponent.

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Negative powers

• A negative power is simply the reciprocal of a positive power.

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Fractional powers

• When a number is raised to a fractional power, the numerator functions like a real exponent, and the denominator functions as the index.

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Raising a power to a power

• When you raise a power to a power you simply multiply the exponents

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The Zero Power

• ANYTHING to the zero power is 1

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The First Power

• ANYTHING to the power of 1 is itself

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Distributing exponents

• When several numbers are inside parenthesis, the exponent outside the parenthesis must be distributed to all of the numbers within.

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But watch out for…

• Exponents are shorthand for multiplication, so the rules apply only when you multiply or divide the same base.

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Radicals

• The square root of a positive number x is the number that when squared, equals x.

• On the ACT you will not have to worry about negative exponents.

• The cubed root of a positive number x is the number that, when cubed, equals x.

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Radicals

• Be sure that you know how to use you calculator more than just the square root and simple exponents.

• The ACT is going to have fractional exponents, negative exponents, and all sorts of weird roots.