fix typos

quarto/basics/numbers_types-II.qmd

@@ -657,7 +657,7 @@ val = sum(2^i for i in 0:10)
 numericq(val)
 ```
 
-The value `1023` is called a bias. The exponent is coded as the binary value as a positve integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits.
+The value `1023` is called a bias. The exponent is coded as the binary value as a positive integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits.
 
 To find the storage for, say, $2^4 + 2^2 + 2^0$ or `00000010101` we would add `1023` or `01111111111` and see:
 

quarto/basics/numbers_types.qmd

@@ -677,7 +677,7 @@ val = sum(2^i for i in 0:10)
 numericq(val)
 ```
 
-The value `1023` is called a bias. The exponent is coded as the binary value as a positve integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits.
+The value `1023` is called a bias. The exponent is coded as the binary value as a positive integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits.
 
 To find the storage for, say, $2^4 + 2^2 + 2^0$ or `00000010101` we would add `1023` or `01111111111` and see:
 

quarto/basics/vectors.qmd

@@ -33,7 +33,7 @@ $$
 
 This formula agrees with Pythagorean's theorem for right triangles.
 
-For $n$-dimensional points, the same formula may be used with adjustements to the notation. Suppose $P = (x_1, x_2, \dots, x_n)$ and $Q = (y_1, y_2, \cdots, y_n)$. then
+For $n$-dimensional points, the same formula may be used with adjustments to the notation. Suppose $P = (x_1, x_2, \dots, x_n)$ and $Q = (y_1, y_2, \cdots, y_n)$. then
 
 $$
 d = \overline{PQ} = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2}.
@@ -56,7 +56,7 @@ A two-dimensional vector has two components, as does a point in the Cartesian pl
 
 Suppose a vector is defined as being between two points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ with $P$ the endpoint, then the vector connecting $P$ to $Q$ would be $\vec{v} = \langle x_2 - x_1, ~ y_2 - y_1 \rangle$.
 
-When $P = (0,0)$, the the point $Q = (x_2, y_2)$ has the same components as the vector $\vec{v} = \langle x_2, ~ y_2 \rangle$ leading to a natural indentification between a point and vector, though they represent different things.
+When $P = (0,0)$, the the point $Q = (x_2, y_2)$ has the same components as the vector $\vec{v} = \langle x_2, ~ y_2 \rangle$ leading to a natural identification between a point and vector, though they represent different things.
 
 
 
@@ -767,7 +767,7 @@ Broadcasting is a widely used and powerful surface syntax which we will employ o
 
 #### Mapping a function over a collection
 
-The `map` function is very much related to broadcasting, in that it applies a function to each element of an iterable. When more than one iterable is specifed, `map` applies the function to the `zip`ped iterables. Unlike broadcasting, `map` does not reshape the underlying iterables.
+The `map` function is very much related to broadcasting, in that it applies a function to each element of an iterable. When more than one iterable is specified, `map` applies the function to the `zip`ped iterables. Unlike broadcasting, `map` does not reshape the underlying iterables.
 
 Similarly named functions are found in many different programming languages, as `map` is one of the foundational higher-order, functional programming operations. (The "dot" broadcast is mostly limited to `Julia` and mirrors a similar usage of a dot in `MATLAB`.) For those familiar with other programming languages, using `map` may seem more natural. Its syntax is `map(f, xs)`. Additional iterables are passed after `xs`.
 
@@ -787,7 +787,7 @@ sum(map(sin, xs))
 
 This has a performance drawback---there are two passes through the container, one to apply `sin` another to add.
 
-For this task, the `sum` reduction, as others, allows a function to be specified that is applied to each value in the container while the sum is being computed. This argument comes first. A recommened alternative to the previous would be:
+For this task, the `sum` reduction, as others, allows a function to be specified that is applied to each value in the container while the sum is being computed. This argument comes first. A recommended alternative to the previous would be:
 
 ```{julia}
 sum(sin, xs)