Unit 1: The Number System
Rational Numbers Review Material
Math 7 Accelerated
The unit 1 test will be on the application of the following skills and concepts:
Rational Numbers
Rational numbers can be expressed as one integer divided by another, where the denominator is not zero.
Rational numbers can be:
-integers
-fractions
-decimals that repeat
-decimals that terminate
Irrational numbers are decimals that go on forever and do not repeat
Dividing by zero:
You cannot divide by zero. Look at the example 13 ÷ 0.
We can rewrite this as multiplication: 0 x _____ = 13
But zero times anything is zero so there is no number that when multiplied by zero will equal 13.
The value of 13 ÷ 0 is undefined.
Expressing fractions as decimals:
Common Fractions:
1
= 0.5
2
1
= 0.25
4
3
= 0.75
4
1
= 0. 3̅
3
2
= 0. 6̅
3
Denominators of Powers of 10:
When possible, determine an equivalent fraction with a power of ten. Then determine how many zeros are in
the denominator and move the decimal to the left that many times.
2
4
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑎𝑛𝑑 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑏𝑦 2.
= 0.4
5
10
4
16
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑎𝑛𝑑 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑏𝑦 4.
= 0.16
25
100
Long Division:
Use long division to express fractions as decimals.
*Remember that the numerator always goes in the house*
The decimals will terminate or repeat. We show repeating decimals by placing a bar above the repeating
number(s).
Expressing Repeating Decimals as Fractions
Step 1: Set the repeating decimal equal to zero
Step 2: Determine how many digits repeat
Step 3: Multiply both sides of the equation by 10 to the
power of how many digits there are
Step 4: Rewrite the equation in expanded notation
Step 5: Replace the decimal with x
Step 6: Solve the equation
x=0.77777
1 digits repeat
10x=10(0.77777)
10x = 7.7777
10x = 7 + 0.77777
10x = 7 + x
9x = 7
x = 7/9
Square Roots
Square Root of Fractions and Decimals
Separate the numerator and denominator and determine the square root of each
√
16
4
√16
=
=
100 √100 10
Turn decimals into fractions and perform the same steps.
Square Root of Large Perfect Square
Use the perfect squares of multiples of tens to determine the square root of larger numbers.
Step 1: Determine which two multiples of ten the number lies between
√1296 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 √900 𝑎𝑛𝑑 √1600 𝑠𝑜 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑖𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 30 𝑎𝑛𝑑 40
There must be a 3 in the tens place
Step 2: Use the digit in the ones place to determine the digit in the ones place of the square root
42 = 16 𝑎𝑛𝑑 62 = 36 𝑠𝑜 𝑡ℎ𝑒 𝑜𝑛𝑒𝑠 𝑝𝑙𝑎𝑐𝑒 𝑖𝑠 𝑒𝑖𝑡ℎ𝑒𝑟 𝑎 4 𝑜𝑟 𝑎 6
Step 3: Determine if 1296 is closer to 900 or 1600.
It is closer to 1600 so use the higher number for the ones place
The answer is 36
Step 4: Check your answer
36*36 =1296
Approximating Non-Perfect Square Roots
Determine which two integers the square root will lie between. Then determine which integer is it closer to and
approximate the square root.
Example: √22 𝑖𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 √16 𝑎𝑛𝑑√25 𝑠𝑜 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 4 𝑎𝑛𝑑 5
22 is closer to 25 so a good approximation would be about 4.7
Absolute Value
Absolute value is the distance from zero. Absolute value must always be positive since it represents a distance.
The absolute value of 2 and -2 is 2 because both are 2 units away from zero.
Opposites
Opposites are numbers that are the same distance from zero. Their sum is always zero.
Opposites cancel each other out. For example, a deposit of $10 and a withdrawal of $10 will cancel each other
out leaving the original amount in the account the same.
Comparing and Ordering Rational Numbers
Positive numbers are larger than negative numbers.
Positive numbers that are further from zero are larger.
Negative numbers that are further from zero are smaller.
> Greater than
Examples: -2 < -1
< Less than
-7 > -20
*Be able to compare and order positive and negative numbers, as well as plot them on the number line.