diff --git a/ch04_keys.adoc b/ch04_keys.adoc
index 6e485ee1..40cfbfb4 100644
--- a/ch04_keys.adoc
+++ b/ch04_keys.adoc
@@ -215,7 +215,7 @@ more complex pattern of dots on a unfathomably large grid.
 .Elliptic curve cryptography: visualizing an elliptic curve over F(p), with p=17
 image::images/mbc3_0403.png["ecc-over-F17-math"]
 
-So, for example, the following is a point P with coordinates (x,y) that
+So, for example, the following is a point P with coordinates (x, y) that
 is a point on the +secp256k1+ curve:
 
 [source, python]
@@ -770,7 +770,7 @@ public keys are known as _compressed public keys_, and the original 65-byte keys
 results in smaller transactions, allowing more payments to be made in the same
 block.
 
-As we saw in the section <<public_key_derivation>>, a public key is a point (x,y) on an
+As we saw in the section <<public_key_derivation>>, a public key is a point (x, y) on an
 elliptic curve. Because the curve expresses a mathematical function, a
 point on the curve represents a solution to the equation and, therefore,
 if we know the _x_ coordinate, we can calculate the _y_ coordinate by