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Math Knowledge

Arithmetic & Number Skills

Arithmetic and number skills are the foundation of Math Knowledge. Many algebra and geometry problems become much easier when you are comfortable working with integers, fractions, decimals, factors, exponents, roots, and common conversions. The goal of this lesson is to make these skills quick and reliable.

Section 1

Integers & Order of Operations

Integers are positive whole numbers, negative whole numbers, and zero. Examples include 8, -5, 0, 17, and -24.

Positive and Negative Numbers

When adding numbers with the same sign, add their values and keep the sign. Example: -4+(-6)=-10.

When adding numbers with different signs, subtract the smaller absolute value from the larger absolute value. Keep the sign of the number with the larger absolute value. Examples: 9+(-4)=5 and 4+(-9)=-5.

Subtracting Negative Numbers

Subtracting a negative number becomes addition. Example: 7-(-3) becomes 7+3=10.

Multiplication and Division Sign Rules

Positive × positive = positive.

Negative × negative = positive.

Positive × negative = negative.

Negative × positive = negative.

The same sign rules apply to division. Examples: (-4)(-3)=12, (-4)(3)=-12, and 12÷(-3)=-4.

Absolute Value

Absolute value is the distance a number is from zero. For example, |5|=5 and |-5|=5. Absolute value is never negative.

Order of Operations

Use PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

Multiplication and division have equal priority, so perform them from left to right. Addition and subtraction also have equal priority, so perform them from left to right.

Rules and formulas

  • Follow PEMDAS.
  • For operations with equal priority, work from left to right.

Example 1

Evaluate 8-3×2.

Solution

  1. 1Multiplication comes before subtraction.
  2. 23×2=6.
  3. 3Now subtract: 8-6=2.
Answer: 2

Why: The multiplication must be completed before the subtraction because multiplication comes earlier in the order of operations.

Example 2

Evaluate -6+11-4.

Solution

  1. 1Addition and subtraction have equal priority, so work from left to right.
  2. 2-6+11=5.
  3. 3Then: 5-4=1.
Answer: 1

Why: When only addition and subtraction remain, solve them in the order they appear from left to right.

Example 3

Evaluate 18÷(5-2)+22.

Solution

  1. 1Start with the parentheses: 5-2=3.
  2. 2Now evaluate the exponent: 22=4.
  3. 3The expression becomes 18÷3+4.
  4. 4Divide: 18÷3=6.
  5. 5Then add: 6+4=10.
Answer: 10

Why: PEMDAS tells us to complete parentheses and exponents before division and addition.

Section 2

Fractions

A fraction represents part of a whole. In 35, 3 is the numerator and 5 is the denominator. The denominator cannot equal zero.

Simplifying Fractions

To simplify a fraction, divide the numerator and denominator by the same common factor.

For 1218, both numbers are divisible by 6: 12÷6=2 and 18÷6=3. Therefore, 1218=23.

Equivalent Fractions

Equivalent fractions have the same value even though they use different numbers. For example, 12=24=36.

Create equivalent fractions by multiplying or dividing the numerator and denominator by the same nonzero number.

Adding and Subtracting Fractions

Fractions need the same denominator before they can be added or subtracted.

If denominators already match, keep the denominator and combine the numerators: 27+37=57.

If denominators differ, first find a common denominator.

Multiplying Fractions

Multiply numerator by numerator and denominator by denominator. Then simplify if possible.

Dividing Fractions

Use KEEP, CHANGE, FLIP: keep the first fraction, change division to multiplication, and flip the second fraction.

For example, 23÷45 becomes 23×54.

Mixed Numbers

A mixed number contains a whole number and a fraction, such as 213.

To convert it to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the original denominator: 2×3=6, 6+1=7, so 213=73.

Rules and formulas

  • Addition and subtraction require a common denominator.
  • Multiply fractions straight across.
  • For division, keep, change, flip.

Example 1

Add 38+14.

Solution

  1. 1The denominators are different.
  2. 2Convert 14 to eighths: 14=28.
  3. 3Now add: 38+28=58.
Answer: 58

Why: Fractions can only be added directly when they represent the same-sized pieces, so the denominators need to match first.

Example 2

Multiply 45×1516.

Solution

  1. 1Multiply the numerators: 4×15=60.
  2. 2Multiply the denominators: 5×16=80.
  3. 3This gives 6080.
  4. 4Both 60 and 80 are divisible by 20: 60÷20=3 and 80÷20=4.
Answer: 34

Why: Fractions are multiplied straight across. The final result should then be reduced to simplest form.

Example 3

Divide 214÷35.

Solution

  1. 1Convert the mixed number: 214=94.
  2. 2The problem is now 94÷35.
  3. 3Use keep, change, flip: 94×53.
  4. 4Multiply: 4512.
  5. 5Simplify: 4512=154.
  6. 6Convert to a mixed number: 15÷4=3 remainder 3, so 154=334.
Answer: 334

Why: Dividing by a fraction is equivalent to multiplying by its reciprocal.

Section 3

Decimals

Decimals represent parts of a whole using place value. Common equivalents include 0.5=12, 0.25=14, and 0.75=34.

Decimal Place Value

In 12.487, 4 is in the tenths place, 8 is in the hundredths place, and 7 is in the thousandths place.

Adding and Subtracting Decimals

Always line up the decimal points. You can add zeros to the end of a decimal without changing its value: 4.7=4.70.

Multiplying Decimals

First multiply as though the decimal points are not present. Then count the total number of decimal places in both original numbers and place that many decimal places in the answer.

Dividing Decimals

If the divisor contains a decimal, move its decimal point until it becomes a whole number. Move the decimal point in the dividend the same number of places, then divide normally.

Common Fraction and Decimal Equivalents

12=0.5, 14=0.25, 34=0.75, 15=0.2, and 110=0.1.

Rules and formulas

  • Line up decimal points for addition and subtraction.
  • Count decimal places after multiplying.
  • Make the divisor a whole number before dividing.

Example 1

Calculate 7.35+2.8.

Solution

  1. 1Line up the decimal points.
  2. 2Rewrite 2.8 as 2.80.
  3. 3Now add: 7.35+2.80=10.15.
Answer: 10.15

Why: Lining up the decimal points makes sure digits with the same place value are added together.

Example 2

Calculate 1.4×0.6.

Solution

  1. 1Ignore the decimal points temporarily: 14×6=84.
  2. 21.4 has one decimal place and 0.6 has one decimal place.
  3. 3That is two decimal places total, so 1.4×0.6=0.84.
Answer: 0.84

Why: When multiplying decimals, the total number of decimal places in the factors determines the decimal placement in the product.

Example 3

Calculate 6.3÷0.9.

Solution

  1. 1Move the decimal in 0.9 one place right to make 9.
  2. 2Move the decimal in 6.3 one place right as well.
  3. 3The problem becomes 63÷9.
  4. 4Now divide: 63÷9=7.
Answer: 7

Why: Moving both decimals the same number of places creates an equivalent division problem while making the divisor a whole number.

Section 4

Factors, Multiples, GCF & LCM

Factors divide evenly into a number. Multiples are created by multiplying a number by whole numbers.

Factors

The factors of 12 are 1, 2, 3, 4, 6, and 12.

Multiples

The multiples of 6 begin 6, 12, 18, 24, 30, 36, and continue.

Prime Numbers

A prime number has exactly two positive factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and 13.

The number 1 is not prime.

Prime Factorization

Prime factorization writes a number as a product of prime numbers.

For 24: 24=2×12, 12=2×6, and 6=2×3. Therefore, 24=2×2×2×3=23×3.

Greatest Common Factor

The Greatest Common Factor, or GCF, is the largest factor shared by two or more numbers.

Least Common Multiple

The Least Common Multiple, or LCM, is the smallest positive multiple shared by two or more numbers.

Rules and formulas

  • GCF means largest shared factor.
  • LCM means smallest shared positive multiple.

Example 1

Find the GCF of 24 and 36.

Solution

  1. 1Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  2. 2Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
  3. 3The shared factors are 1, 2, 3, 4, 6, and 12.
  4. 4The greatest shared factor is 12.
Answer: 12

Why: The GCF is the largest number that divides evenly into both original numbers.

Example 2

Find the LCM of 8 and 12.

Solution

  1. 1Multiples of 8: 8, 16, 24, 32, 40, and so on.
  2. 2Multiples of 12: 12, 24, 36, 48, and so on.
  3. 3The first shared positive multiple is 24.
Answer: 24

Why: The LCM is the smallest positive number that is a multiple of both original numbers.

Example 3

Find the prime factorization of 60.

Solution

  1. 1Break 60 into factors: 60=6×10.
  2. 2Break both into primes: 6=2×3 and 10=2×5.
  3. 3Therefore, 60=2×3×2×5.
  4. 4Rearrange: 60=22×3×5.
Answer: 22×3×5

Why: Prime factorization continues breaking a number into factors until every factor remaining is prime.

Section 5

Exponents & Roots

An exponent tells you how many times a base is multiplied by itself. For example, 53 means 5×5×5=125. Here, 5 is the base and 3 is the exponent.

Squares and Cubes

A number raised to the second power is squared: 62=36.

A number raised to the third power is cubed: 43=64.

Multiplying Powers With the Same Base

When multiplying powers with the same base, add the exponents: x2× x3=x5.

Dividing Powers With the Same Base

When dividing powers with the same base, subtract the exponents: x7x3=x4.

Power of a Power

When raising a power to another power, multiply the exponents: (x2)3=x6.

Zero Exponent

Any nonzero number raised to the zero power equals 1. For example, 70=1.

Square Roots

A square root asks, “What number multiplied by itself gives this value?” For example, √49=7 because 7×7=49.

Cube Roots

A cube root asks, “What number multiplied by itself three times gives this value?” For example, ∛27=3 because 3×3×3=27.

Common Perfect Squares

12=1, 22=4, 32=9, 42=16, 52=25, 62=36, 72=49, 82=64, 92=81, 102=100, 112=121, and 122=144.

Rules and formulas

  • Same-base multiplication: add exponents.
  • Same-base division: subtract exponents.
  • Power of a power: multiply exponents.
  • A nonzero number to the zero power is 1.

Example 1

Simplify 32×34.

Solution

  1. 1The bases are the same.
  2. 2When multiplying powers with the same base, add the exponents: 2+4=6.
  3. 3Therefore, 32×34=36.
  4. 4If evaluated, 36=729.
Answer: 36, or 729

Why: Multiplying powers with the same base combines all of the repeated factors, so the exponents are added.

Example 2

Evaluate √144-23.

Solution

  1. 1Evaluate the square root: √144=12.
  2. 2Evaluate the exponent: 23=8.
  3. 3Now subtract: 12-8=4.
Answer: 4

Why: Roots and exponents are evaluated before subtraction according to the order of operations.

Example 3

Simplify (x3)2x4.

Solution

  1. 1Start with the power of a power: 3×2=6, so (x3)2=x6.
  2. 2Now divide: x6x4.
  3. 3Subtract the exponents: 6-4=2.
  4. 4Therefore, the result is x2.
Answer: x2

Why: A power raised to another power multiplies exponents, while division of powers with the same base subtracts exponents.

Section 6

Scientific Notation & Basic Conversions

Scientific notation writes very large or very small numbers in the form a×10n, where 1≤ a<10. The first number must be at least 1 but less than 10.

Large Numbers

For 48,000, move the decimal until the first number is between 1 and 10: 48,000 becomes 4.8. The decimal moved 4 places left, so 48,000=4.8×104.

Large numbers normally use positive exponents.

Small Numbers

For 0.0007, move the decimal until the first number is between 1 and 10: 0.0007 becomes 7. The decimal moved 4 places right, so 0.0007=7×10-4.

Small decimals normally use negative exponents.

Scientific Notation to Standard Form

For a positive exponent, move the decimal right: 3.2×103=3,200.

For a negative exponent, move the decimal left: 4.5×10-3=0.0045.

Basic Conversions

12 inches = 1 foot; 3 feet = 1 yard; 36 inches = 1 yard; 16 ounces = 1 pound; 60 seconds = 1 minute; 60 minutes = 1 hour; 24 hours = 1 day.

Choosing Multiply or Divide

Going from a larger unit to a smaller unit usually means multiplying, such as feet to inches.

Going from a smaller unit to a larger unit usually means dividing, such as inches to feet. Think about how many smaller units fit inside one larger unit.

Rules and formulas

  • Scientific notation starts with a number at least 1 but less than 10.
  • Positive powers of 10 move the decimal right; negative powers move it left.

Example 1

Write 620,000 in scientific notation.

Solution

  1. 1Move the decimal until the leading number is between 1 and 10: 620,000 becomes 6.2.
  2. 2The decimal moved 5 places left.
  3. 3Therefore, 620,000=6.2×105.
Answer: 6.2×105

Why: The positive exponent shows how many places the decimal must move to recreate the original large number.

Example 2

Write 3.7×10-4 in standard form.

Solution

  1. 1The exponent is -4.
  2. 2Move the decimal 4 places left: 3.7 becomes 0.37, then 0.037, then 0.0037, then 0.00037.
Answer: 0.00037

Why: A negative exponent represents a number smaller than 1, so the decimal moves left.

Example 3

A board is 7 feet long. How many inches long is it?

Solution

  1. 1Use 1 foot = 12 inches.
  2. 2There are 7 feet.
  3. 3Multiply: 7×12=84.
Answer: 84 inches

Why: We are converting from a larger unit, feet, to a smaller unit, inches, so we multiply by the number of inches in each foot.

Quick Review

Integers & Order of Operations

  • Know positive and negative sign rules.
  • Subtracting a negative becomes addition.
  • Use absolute value as distance from zero.
  • Follow PEMDAS.
  • Work left to right for equal-priority operations.

Fractions

  • Simplify when possible.
  • Addition and subtraction require a common denominator.
  • Multiply straight across.
  • Divide using keep, change, flip.
  • Convert mixed numbers to improper fractions when necessary.

Decimals

  • Line up decimal points when adding or subtracting.
  • Count decimal places when multiplying.
  • Make the divisor a whole number when dividing.
  • Know common fraction-decimal equivalents.

Factors & Multiples

  • GCF = largest shared factor.
  • LCM = smallest shared positive multiple.
  • Prime factorization uses only prime factors.
  • 1 is not prime.

Exponents & Roots

  • Same-base multiplication: add exponents.
  • Same-base division: subtract exponents.
  • Power of a power: multiply exponents.
  • A nonzero number to the zero power equals 1.
  • Square roots undo squares.

Scientific Notation & Conversions

  • Scientific notation begins with a number at least 1 but less than 10.
  • Large numbers generally use positive exponents.
  • Small decimals generally use negative exponents.
  • Larger unit to smaller unit usually means multiply.
  • Smaller unit to larger unit usually means divide.